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CHAPTER VII. FURTHER APPLICATIONS
In this chapter we discuss some further geometric problems about di_erent
types of natural operators. First we deduce that all natural bilinear operators
transforming a vector _eld and a di_erential k-form into a di_erential k-form
form a 2-parameter family. This further clari_es the well known relation between
Lie derivatives and exterior derivatives of k-forms. From the technical
point of view this problem can be considered as a preparatory exercise to the
problem of _nding all bilinear natural operators of the type of the Frolicher-
Nijenhuis bracket. We deduce that in general case all such operators form a
10-parameter family. Then we prove that there is exactly one natural operator
transforming general connections on a _bered manifold Y ! M into general connections
on its vertical tangent bundle V Y ! M. Furthermore, starting from
some geometric problems in analytical mechanics, we deduce that all _rst-order
natural operators transforming second-order di_erential equations on a manifold
M into general connections on its tangent bundle TM ! M form a one parameter
family. Further we study the natural transformations of the jet functors.
The construction of the bundle of all r-jets between any two manifolds can be
interpreted as a functor Jr on the product category Mfm _Mf. We deduce
that for r _ 2 the only natural transformations of Jr into itself are the identity
and the contraction, while for r = 1 we have a one-parameter family of homotheties.
This implies easily that the only natural transformation of the functor
of the r-th jet prolongation of _bered manifolds into itself is the identity. For
the second iterated jet prolongation J1(J1Y ) of a _bered manifold Y we look
for an analogy of the canonical involution on the second iterated tangent bundle
TTM. We prove that such an exchange map depends on a linear connection on
the base manifold and we give a simple list of all natural transformations of this
type.
The next section is devoted to some problems from Riemannian geometry.
Here we complete our study of natural connections on Riemannian manifolds,
we prove the Gilkey theorem on natural di_erential forms and we _nd all natural
lifts of Riemannian metrics to the tangent bundles. We also deduce that all
natural operators transforming linear symmetric connections into exterior forms
are generated by the Chern forms. Since there are no natural forms of odd
degree, all of them are closed.
In the last section, we present a survey of some results concerning the multilinear
natural operators which are based heavily on the (linear) representation
theory of Lie algebras. First we treat the naturality over the whole category
Mfm, where the main tools come from the representation theory of in_nite dimensional
algebras of vector _elds. At the very end we comment briey on the
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
250 Chapter VII. Further applications
category of conformal (Riemannian) manifolds, which leads to _nite dimensional
representation theory of some parabolic subalgebras of the Lie algebras of the
pseudo orthogonal groups.
30. The Frolicher-Nijenhuis bracket
The main goal of this section is to determine all bilinear natural operators of
the type of the Frolicher-Nijenhuis bracket. But we _nd it useful to start with a
technically simpler problem, which can serve as an introduction.
30.1. Bilinear natural operators T __pT_ _pT_. We are going to study
the natural operators transforming a vector _eld and an exterior p-form into an
exterior p-form. In order to get results of geometric interest, it is reasonable to
restrict ourselves to the bilinear operators. The two simplest examples of such
operators are (X; !) 7! diX! and (X; !) 7! iXd!.
Proposition. All bilinear natural operators T _ _pT_ _pT_ form the 2-
parameter family
(1) k1diX! + k2iXd!; k1; k2 2 R:
Proof. First of all, every such operator has _nite order r by the bilinear Peetre
theorem. The canonical coordinates on the standard _ber S = Jr
0TRm _
Jr
0_pT_Rm are Xi_ , bi1:::ip;_, j_j _ r, j_j _ r, while the canonical coordinates
on the standard _ber Z = _pRm_ are ci1:::ip . Since we consider the bilinear
operators, even the associated maps f : S ! Z are bilinear in Xi_ and bi1:::ip;_.
Using the homotheties in GL(m) _ Gr+1
m , we obtain
(2) kpf(Xi_ ; bi1:::ip;_) = f(kj_j1Xi_; kp+j_jbi1:::ip;_):
This implies that only the products Xibi1:::ip;j and Xi
jbi1:::ip can appear in f.
(In particular, every natural bilinear operator T __pT_ _pT_ is a _rst order
operator.) Denote by f = f1 + f2 the corresponding decomposition of f.
The transformation laws of bi1:::ip , bi1:::ip;j can be found in 25.4 and one
deduces easily
(3) _X i = ai
jXj ; _X i
j = ai
kl~akj
Xl + ai
kXk
l ~al
j :
In particular, the transformation laws with respect to the subgroup GL(m) _
G2
m are tensorial in all cases. Hence we _rst have to determine the GL(m)-
equivariant bilinear maps Rm__pRm_Rm_ ! _pRm_. Consider the following
diagram
(4)
Rm _ _pRm_ Rm_ w
f1
u
id _ Altp id
z
u
_pRm_
y
u
u
Altp
Rm _ p+1Rm_ wpRm_
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
30. The Frolicher-Nijenhuis bracket 251
where Alt denotes the alternator of the indicated degree. The vertical maps are
also GL(m)-equivariant and the GL(m)-equivariant map in the bottom row can
be determined by the invariant tensor theorem. This implies that f1 is a linear
combination of the contraction of Xi with the derivation entry in bi1:::ip;j and
of the contraction of Xi with a non-derivation entry in bi1:::ip;j followed by the
alternation. To specify f2, consider the diagram
(5)
Rm Rm_ _pRm_ w
~ f2
u
id id Altp
z
u
_pRm_
y
u
u
Altp
Rm _ p+1Rm_ wpRm_
where ~ f2 is the linearization of f2. Taking into account the maps in the bottom
row determined by the invariant tensor theorem, we conclude similarly as
above that f2 is a linear combination of the inner contraction Xj
j multiplied
by bi1:::ip and of the contraction Xj
i1bi2:::ipj followed by the alternation. Thus,
the equivariance of f with respect to GL(m) leads to the following 4-parameter
family
(6) fi1:::ip = aXjbi1:::ip;j + bXjbj[i2:::ip;i1] + cXj
j bi1:::ip + eXj
[i1
bi2:::ip]j
a, b, c, e 2 R.
The equivariance of f on the kernel ai
j = _ij
is expressed by the relation
(7)
0 = aXj(bki2:::ipaki
1j + _ _ _ + bi1:::ip1kaki
pj)+
bXjbk[i2:::ipaki
1]j + cakk
jXjbi1:::ip + eXjakj
[i1bi2:::ip]k:
This implies
(8) c = 0 and a = b + e
which gives the coordinate form of (1). _
30.2. The Lie derivative. Proposition 30.1 gives a new look at the well known
formula expressing the Lie derivative LX! of a p-form as the sum of diX! and
iXd!. Clearly, the Lie derivative operator on p-forms (X; !) 7! LX! is a bilinear
natural operator T _ _pT_ _pT_. By proposition 30.1, there exist certain
real numbers a1 and a2 such that
LX! = a1diX! + a2iXd!
for every vector _eld X and every p-form ! on m-manifolds. If we evaluate
a1 = 1 = a2 in two suitable special cases, we obtain an interesting proof of the
classical formula.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
252 Chapter VII. Further applications
30.3. Bilinear natural operators T__pT_ _qT_. These operators can be
determined in the same way as in 30.1, see [Kol_a_r, 90b]. That is why we restrict
ourselves to the result. The only natural bilinear operators T __pT_ _p1T_
or _p+1T_ are the constant multiples of iX! or d(iXd!), respectively. In the
case q 6= p 1, p, p + 1, we have the zero operator only.
30.4. Bilinear natural operators of the Frolicher-Nijenhuis type. The
wedge product of a di_erential q-form and a vector valued p-form is a bilinear
map q(M) _ p(M; TM) ! p+q(M; TM) characterized by ! ^ ('
X) = (! ^ ') X for all ! 2 q(M), ' 2 p(M), X 2 X(M). Further let
C : p(M; TM) ! p1(M) be the contraction operator de_ned by C(!X) =
i(X)! for all ! 2 p(M), X 2 X(M). In particular, for P 2 0(M; TM) we have
C(P) = 0. Clearly C(i(P)Q) is a linear combination of C(i(Q)P), i(P)(C(Q)),
i(Q)(C(P)), P 2 p(M; TM), Q 2 q(M; TM). By I we denote IdTM, viewed
as an element of 1(M; TM).
Theorem. For dimM _ p + q, all bilinear natural operators A: p(M; TM) _
q(M; TM) ! p+q(M; TM) form a vector space linearly generated by the
following 10 operators
(1)
[P;Q]; dC(P) ^ Q; dC(Q) ^ P; dC(P) ^ C(Q) ^ I;
dC(Q) ^ C(P) ^ I; dC(i(P)Q) ^ I; i(P)dC(Q) ^ I;
i(Q)dC(P) ^ I; d(i(P)C(Q)) ^ I; d(i(Q)C(P)) ^ I:
These operators form a basis if p, q _ 2 and m _ p + q + 1.
30.5. Remark. If p or q is _ 1, then all bilinear natural operators in question
are generated by those terms from 30.4.(1) that make sense. For example, in the
extreme case p = q = 0 our result reads that the only bilinear natural operators
X(M)_X(M) ! X(M) are the constant multiples of the Lie bracket. This was
proved by [van Strien, 80], [Krupka, Mikol_a_sov_a, 84], and in an `in_nitesimal'
sense by [de Wilde, Lecomte, 82]. For a detailed discussion of all special cases
we refer the reader to [Cap, 90]. Clearly, for m < p+q we have the zero operator
only.
30.6. To prove theorem 30.4, we start with the fact that the bilinear Peetre
theorem implies that every A has _nite order r. Denote by Pi
j1:::jp or Qi
j1:::jq
the canonical coordinates on Rm _pRm_ or Rm _qRm_, respectively. The
associated map A0 of A is bilinear in P's and Q's and their partial derivatives up
to order r. Using equivariance with respect to homotheties in GL(m), we _nd
that A0 contains only the products Pi
j1:::jp;kQmn
1:::nq and Qi
j1:::jq;kPm
n1:::np , where
the _rst term in both expressions means the partial derivative with respect to
xk. In other words, A is a _rst order operator and A0 is a sum A1 + A2 where
A1 : Rm _pRm_ Rm_ _ Rm _qRm_ ! Rm _p+qRm_
A2 : Rm _pRm_ _ Rm _qRm_ Rm_ ! Rm _p+qRm_
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
30. The Frolicher-Nijenhuis bracket 253
are bilinear maps. One _nds easily that the transformation law of Pi
j1:::jp;k is
(1)
_ Pi
j1:::jp;k = Plm
1:::mp;nail
~am1
j1 : : : ~amp
jp
~ank
+ Plm
1:::mp (ai
ln~am1
j1 : : : ~amp
jp
~ank
+ ail
~am1
j1k~am2
j2 : : : ~amp
jp
+ : : :
+ ail
~am1
j1 : : : ~amp
jpk):
30.7. Taking into account the canonical inclusion GL(m) _ G2
m, we see that
the linear maps associated with the bilinear maps A1 and A2, which will be
denoted by the same symbol, are GL(m)-equivariant. Consider _rst the following
diagram
(1)
Rm _pRm_ Rm_ Rm _qRm_ w
A1
u
id Altp id Altq
z
u
Rm _p+qRm_
y
u
u
id Altp+q
Rm pRm_ Rm_ Rm qRm_ wRm p+qRm_
where Alt denotes the alternator of the indicated degree. It su_ces to determine
all equivariant maps in the bottom row, to restrict them and to take the alternator
of the result. By the invariant tensor theorem, all GL(m)-equivariant maps
2Rm p+q+1Rm_ ! Rm p+qRm_ are given by all kinds of permutations
of the indices, all contractions and tensorizing with the identity. Since we apply
this to alternating forms and use the alternator on the result, permutations do
not play a role.
In what follows we discuss the case p _ 2, q _ 2 only and we leave the other
cases to the reader. (A direct discussion shows that in the remaining cases the
list (2) below should be reduced by those terms that do not make sense, but
the next procedure leads to theorem 30.4 as well.) Constructing A1, we may
contract the vector _eld part of P into a non-derivation entry of P or into the
derivation entry of P or into Q, and we may contract the vector _eld part of
Q into Q or into a non-derivation entry of P or into the derivation entry of
P, and then tensorize with the identity of Rm. This gives 8 possibilities. If
we perform only one contraction, we get 6 further possibilities, so that we have
a 14-parameter family denoted by the lower case letters in the list (2) below.
Constructing A2, we obtain analogously another 14-parameter family denoted
by upper case letters in the list (2) below. Hence GL(m)-equivariance yields the
following expression for A0 (we do not indicate alternation in the subscripts and
we write _, _ for any kind of free form-index on the right hand side)
(2)
aPm
m_;kQn
n__i
l + bPm
_;mQn
n__i
l + cPm
_;kQn
nm__i
l + dPm
mn_;kQn_ _i
l+
ePm
n_;mQn_ _i
l + fPm
n_;kQn
m__i
l + gPm
m_;nQn__i
l + hPm
_;nQn
m__i
l+
iPm
m_;kQi
_ + jPm
_;mQi
_ + kPm
_;kQi
m_ + lPi
_;kQn
n_ + mPi
n_;kQn_ +
nPi
_;nQn_ + APm
m_Qn
n_;k_i
l + BPm
m_Qn_;n_i
l + CPm
mn_Qn_;k_i
l+
DPm
_ Qn
nm_;k_i
l + EPm
_ Qn
m_;n_i
l + FPm
n_Qn
m_;k_i
l + GPm
_ Qn
n_;m_i
l+
HPm
n_Qn_;m_i
l + IPi_ Qn
n_;k + JPi_
Qn_;n + KPi
n_Qn_;k+
LPm
m_Qi
_;k +MPm
_ Qi
m_;k + NPm
_ Qi
_;m:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
254 Chapter VII. Further applications
30.8. Then we consider the kernel K of the jet projection G2
m
! G1
m. Using
30.5.(1) with ai
j = _ij
, we evaluate that A0 is K-equivariant if and only if the
following coordinate expression
(1)_
BPm
m_Qt
_an
tn + (1)qqBPm
m_Qn
t_at
nk + bPt_Qn
n_am
tm
(1)p+qpbPm
t_Qn
n_at
mk + ((1)qc D (1)q(q 1)G)Pm
_ Qn
nt_at
mk+
(C (1)qd (1)p+q(p 1)g)Pm
mt_Qn_ at
nk + ePm
n_Qn_ at
mt+
(H e)Pm
t_Qn_ at
mn + EPm
_ Qt
m_an
tn + (h E)Pm
_ Qn
t_at
mn+
((1)qqH (1)qf F)Pm
n_Qn
t_at
mk+
(F + (1)qf (1)p+qph)Pm
n_Qt
m_an
tk
(1)p+q(p 1)ePm
nt_Qn_at
mk
(1)q(q 1)EPm
_ Qn
mt_at
nk
_
_i
l + jPt_ Qi
_am
tm + (1)p+qpjPm
t_Qi
_at
mk+
JPi_ Qt
_am
tm + (1)qqJPi_Qm
t_at
mk
((1)qk (1)qqN +M)Pm
_ Qi
t_at
mk+
(K + (1)p+qpn (1)qm)Pi
m_Qt
_am
tk
l(1)qPt_Qm
m_ai
tk+
LPm
m_Qt
_ai
tk
(1)qmPt
m_Qm_ ai
tk +MPm
_ Qt
m_ai
tk + (n + N)Pm
_ Qt
_ai
mt
represents the zero map Rm _pRm_ _ Rm _qRm_ _ Rm S2Rm_ ! Rm
_p+qRm_.
For dimM _ p + q + 1, the individual terms in (1) are linearly independent.
Hence (1) is the zero map if and only if all the coe_cients vanish. This leads to
the following equations
(2)
b = B = e = E = h = H = j = J = l = L = m = M = 0
c = (q 1)G + (1)qD; C = (1)qd + (1)p+q(p 1)g
F = (1)q1f; k = qn; K = (1)p+q1pn; N = n
while a, A, d, D, f, g, G, n, i, I are independent parameters. This yields the
coordinate form of 30.4.(1).
In the case m = p+q, p, q _ 2, there are certain linear relations between the
individual terms of (1). They are described explicitly in [Cap, 90]. But even in
this case we obtain the _nal result in the form indicated in theorem 30.4. _
30.9. Linear and bilinear natural operators on vector valued forms.
Roughly speaking, we can characterize theorem 30.4 by saying that the Frolicher-
Nijenhuis bracket is the only non-trivial operator in the list 30.4.(1), since the
remaining terms can easily be constructed by means of tensor algebra and exterior
di_erentiation. We remark that the natural operators on vector valued
forms were systematically studied by A. Cap. He deduced the complete list of
all linear natural operators p(M; TM) ! q(M; TM) and all bilinear natural
operators p(M; TM) _ q(M; TM) ! r(M; TM), which can be found
in [Cap, 90]. From a general point of view, the situation is analogous to 30.4:
except the Frolicher-Nijenhuis bracket, all other operators in question can easily
be constructed by means of tensor algebra and exterior di_erentiation.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
31. Two problems on general connections 255
30.10. Remark on the Schouten-Nijenhuis bracket. This is a bilinear
operator C1_pTM _ C1_qTM ! C1_p+q1TM introduced geometrically
by [Schouten, 40] and further studied by [Nijenhuis, 55]. In [Michor, 87b] the
natural operators of this type are studied. The problem is technically much
simpler than in the Frolicher-Nijenhuis case and the same holds for the result:
The only natural bilinear operators _pT _ _qT _p+q1T are the constant
multiples of the Schouten-Nijenhuis bracket.
31. Two problems on general connections
31.1. Vertical prolongation of connections. Consider a connection : Y !
J1Y on a _bered manifold Y ! M. If we apply the vertical tangent functor V ,
we obtain a map V : V Y ! V J1Y . Let iY : V J1Y ! J1V Y be the canonical
involution, see 39.8. Then the composition
(1) VY := iY _ V : V Y ! J1V Y
is a connection on V Y ! M, which will be called the vertical prolongation of .
Since this construction has geometrical character, V is an operator J1 J1V
natural on the category FMm;n.
Proposition. The vertical prolongation V is the only natural operator J1
J1V .
We start the proof with _nding the equations of V. If
(2) dyp = Fp
i (x; y)dxi
is the coordinate form of and Y p = dyp are the additional coordinates on V Y ,
then (1) implies that the equations of V are (2) and
(3) dY p = @Fp
i
@yq Y qdxi:
31.2. The standard _ber of V on the category FMm;n is Rn. Let S1 =
J1
0 (J1(Rm _ Rn ! Rm) ! Rm _ Rn) and Z = J1
0 (V (Rm _ Rn) ! Rm),
0 2 Rm _ Rn. By 18.19, the _rst order natural operators are in bijection with
G2
m;n-maps S1 _Rn ! Z over the identity of Rn. The canonical coordinates on
S1 are yp
i , yp
iq, yp
ij and the action of G2
m;n can be found in 27.3. The action of
G2
m;n on Rn is
(1) _ Y p = apq
Y q:
The coordinates on Z are Y p, zp
i = @yp=@xi, Y p
i = @Y p=@xi. By standard
evaluation we _nd the following action of G2
m;n
(2)
_ Y p = apq
Y q; _zp
i = apq
zq
j ~aj
i + ap
j ~aj
i
_ Y p
i = apq
Y q
j ~aj
i + apq
rzr
j Y q~aj
i + ap
qjY q~aj
i :
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
256 Chapter VII. Further applications
The coordinate form of any map S1_Rn ! Z over the identity of Rn is Y p = Y p
and
zp
i = fp
i (Y q; yr
j ; ysk
t; yu
lm)
Y p
i = gp
i (Y q; yr
j ; ysk
t; yu
lm):
First we discuss fp
i . The equivariance with respect to base homotheties yields
(3) kfp
i = fp
i (Y q; kyr
j ; kysk
t; k2yu
lm):
By the homogeneous function theorem, if we _x Y q, then fp
i is linear in yr
j , ysk
t
and independent of yu
lm. The _ber homotheties then give
(4) kfp
i = fp
i (kY q; kyr
j ; ysk
t):
By (3) and (4), fp
i is a sum of an expression linear in yp
i and bilinear in Y p and
yp
iq. Since fp
i is GL(m) _ GL(n)-equivariant, the generalized invariant tensor
theorem implies it has the following form
(5) ayp
i + bY pyq
qi + cY qyp
qi:
The equivariance on the kernel K of the projection G2
m;n
! G1
m
_ G1
n yields
(6) ap
i = aap
i + bY p(aq
qi + aqq
ryr
i ) + cY q(ap
qi + apq
ryr
i ):
This implies a = 1, b = c = 0, which corresponds to 31.1.(2).
For gp
i , the above procedure leads to the same form (5). Then the equivariance
with respect to K yields a = b = 0, c = 1. This corresponds to 31.1.(3). Thus
we have proved that V is the only _rst order natural operator J1 J1V .
31.3. By 23.7, every natural operator A: J1 J1V has a _nite order r. Let f =
(fp
i ; gp
i ) : Sr _Rn ! Z be the associated map of A, where Sr = Jr
0 (J1(Rm+n !
Rm) ! Rm+n). Consider _rst fp
i (Y q; yr
j__) with the same notation as in the
second step of the proof of proposition 27.3. The base homotheties yield
(1) kfp
i = fp
i (Y q; kj_j+1yr
j__):
By the homogeneous function theorem, if we _x Y p, then fp
i are independent
of yp
i__ with j_j _ 1 and linear in yp
i_. Hence the only s-th order term is
's = 'pj_
iq (Y r)yq
j_, j_j = s, s _ 2. Using _ber homotheties we _nd that 's is of
degree s in Y p. Then the generalized invariant tensor theorem implies that 's
is of the form
asyp
iq1:::qs
Y q1 : : : Y qs + bsY pyq1
iq1q2:::qs
Y q2 : : : Y qs :
Consider the equivariance with respect to the kernel of the jet projection Gr+1
m;n
!
Gr
m;n. Using induction we deduce the transformation law
_yp
iq1:::qr
= yp
iq1:::qr
+ ap
tq1:::qryt
i + ap
iq1:::qr
;
while the lower order terms remain unchanged. By direct evaluation we _nd
ar = br = 0. The same procedure takes place for gp
i . Hence A is an operator
of order r 1. By recurrence we conclude A is a _rst order operator. This
completes the proof of proposition 31.1.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
31. Two problems on general connections 257
31.4. Remark. In [Kol_a_r, 81a] it was clari_ed geometrically that the vertical
prolongation V plays an important role in the theory of the original connection
. The uniqueness of V proved in proposition 31.1 gives a theoretical
justi_cation of this phenomenon.
It is remarkable that there is another construction of V using ow prolongations
of vector _elds, see 45.4. The equivalence of both de_nitions is an
interesting consequence of the uniqueness of operator V.
31.5. Natural operators transforming second order di_erential equations
into general connections. We recall that a second order di_erential
equation on a manifold M is usually de_ned as a vector _eld _ : TM ! TTM on
TM satisfying TpM _ _ = idTM, where pM : TM ! M is the bundle projection.
Let LM be the Liouville vector _eld on TM, i.e. the vector _eld generated by
the homotheties. If [_;LM] = _, then _ is said to be a spray. There is a classical
bijection between sprays and linear symmetric connections, which is used in
several branches of di_erential geometry. (We shall obtain it as a special case of
a more general construction.)
A. Dekr_et, [Dekr_et, 88], studied the problem whether an arbitrary second
order di_erential equation on M determines a general connection on TM by
means of the naturality approach. He deduced rather quickly a simple analytical
expression for all _rst order natural operators. Only then he looked for the
geometrical interpretation. Keeping the style of this book, we _rst present the
geometrical construction and then we discuss the naturality problem.
According to 9.3, the horizontal projection of a connection on an arbitrary
_bered manifold Y is a vector valued 1-form on Y , which will be called the
horizontal form of .
On the tangent bundle TM, we have the following natural tensor _eld VM of
type
1
1
_
. Since TM is a vector bundle, its vertical tangent bundle is identi_ed
with TM _ TM. For every B 2 TTM we de_ne
(1) VM(B) = (pTMB; TpMB):
(A general approach to natural tensor _elds of type
1
1
_
on an arbitrary Weil
bundle is explained in [Kol_a_r, Modugno, 92].)
Given a second order di_erential equation _ on M, the Lie derivative L_VM
is a vector valued 1-form on TM. Let 1TTM be the identity on TTM. The
following result gives a construction of a general connection on TM determined
by _.
Lemma. For every second order di_erential equation _ on M, 1
2 (1TTM L_VM)
is the horizontal form of a connection on TM.
Proof. Let xi be local coordinates on M and yi = dxi be the induced coordinates
on TM. The coordinate expression of the horizontal form of a connection on
TM is
(2) dxi
@
@xi + Fj
i (x; y)dxi
@
@yj :
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
258 Chapter VII. Further applications
By (1), the coordinate expression of VM is
(3) dxi
@
@yi :
Having a second order di_erential equation _ of the form
(4) yi @
@xi + _i(x; y) @
@yi
we evaluate directly for L_VM
(5) dxi
@
@xi
@_i
@yj dxj
@
@yi + dyi
@
@yi :
Hence 1
2 (1TM L_VM) has the required form
(6) dxi
@
@xi +
1
2
@_i
@yj dxj
@
@yi : _
31.6. Denote by A the operator from lemma 31.5. By the general theory, the
di_erence of two general connections on TM ! M is a section TM ! V TM
T_M = (TM _ TM) T_M. The identity tensor of TM T_M determines a
natural section IM : TM ! V TM T_M. Hence A + kI is a natural operator
for every k 2 R.
Proposition. All _rst order natural operators transforming second order differential
equations on a manifold into connections on the tangent bundle form
the one-parameter family
A + kI; k 2 R:
Proof. We have the case of a morphism operator from 18.17 with C = Mfm,
F1 = G1 = T, q = id, F2 = T2
1 , G2 = J1T and the additional conditions
that we consider the sections of T2
1
! T and J1T ! T. Let S be the _ber of
J1(T2
1 Rm ! TRm) over 0 2 Rm and Z be the _ber of J1TRm over 0 2 Rm. By
18.19 we have to determine all G3
m-equivariant maps f : S ! Z over the identity
of T0Rm.
Denote by Xi = dxi
dt , Y i = d2xi
dt2 the induced coordinates on T2
1 Rm and by
Xi
j = @Y i=@xj , Y i
j = @Y i=@Xj the induced coordinates on S. By direct evaluation,
one _nds the following action of G3
m
_X
i = ai
jXj ; _ Y i = ai
jkXjXk + ai
(1) jY j
_X
i
j = ai
klm~amj
XkXl + ai
kl~al
jY k + 2ai
kl~ak
mjamn
XlXn + ai
kXk
l ~al
j(2) +
ai
k~al
mjamn
XnY k
l
_ Y i
j = 2ai
kl~al
jXk + ai
kY k
l ~al
j (3)
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
32. Jet functors 259
The standard coordinates Zi, Zi
j on Z have the transformation law
(4) _ Zi = ai
jZj ; _ Zi
j = ai
kl~akj
Zl + ai
kZk
l ~al
j :
Let Zi = Xi and Zi
j = fi
j (Xp; Y q;Xm
n ; Y k
l ) be the coordinate expression of
f. The equivariance with respect to the homotheties in GL(m) _ G3
m yields
(5) fi
j (Xp; Y q;Xm
n ; Y k
l ) = fi
j (kXp; kY q;Xm
n ; Y k
l ):
Hence fi
j do not depend on Xp and Y q. Let ai
j = _ij
, ai
jk = 0. Then the
equivariance condition reads
(6) fi
j (Xm
n ; Y k
l ) = fi
j (Xm
n + am
npqXpXq; Y k
l ):
This implies fi
j are independent of Xm
n . Putting ai
j = _ij
, we obtain
(7) fi
j (Y k
l ) + ai
mjXm = fi
j (Y k
l + 2ak
lmXm)
with arbitrary ai
jk. Di_erentiating with respect to Y k
l , we _nd @fi
j=@Y k
l = const.
Hence fi
j are a_ne functions. By the Invariant tensor theorem, we deduce
(8) fi
j = k1Y i
j + k2_ij
Y k
k + k3_ij
:
Using (7) once again, we obtain k1 = 1
2 , k2 = 0. This gives the coordinate form
of our assertion. _
31.7. Remark. If X is a spray, then the operator A from lemma 31.5 determines
the classical linear symmetric connection induced by X. Indeed, 31.5.(4)
satis_es the spray condition if and only if
@_i
@yj yj = 2_i:
This kind of homogeneity implies _i = bi
jk(x)yjyk. Then the coordinate form of
A(X) is
dyi = bi
jk(x)yjdxk:
32. Jet functors
32.1. By 12.4, the construction of r-jets of smooth maps can be viewed as
a bundle functor Jr on the product category Mfm _Mf. We are going to
determine all natural transformations of Jr into itself. Denote by ^y : M ! N
the constant map of M into y 2 N. Obviously, the assignment X 7! jr
_X
d_X
is a natural transformation of Jr into itself called the contraction. For r = 1,
J1(M;N) coincides with Hom(TM; TN), which is a vector bundle over M _N.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
260 Chapter VII. Further applications
Proposition. For r _ 2 the only natural transformations Jr ! Jr are the
identity and the contraction. For r = 1, all natural transformations J1 ! J1
form the one-parametric family of homotheties X 7! cX, c 2 R.
Proof. Consider _rst the subcategoryMfm_Mfn _Mfm_Mf. The standard
_ber S = Jr
0 (Rm;Rn)0 is a Gr
m
_Gr
n-space and the action of (A;B) 2 Gr
m
_Gr
n
on X 2 S is given by the jet composition
(1) _X = B _ X _ A1:
According to the general theory, the natural transformations Jr ! Jr are in
bijection with the Gr
m
_ Gr
n-equivariant maps f : S ! S.
Write A1 = (~ai
j ; : : : ; ~ai
j1:::jr ), B = (bpq
; : : : ; bpq
1:::qr ), X = (Xp
i ; : : : ;Xp
i1:::ir
) =
(X1; : : : ;Xr). Consider the equivariance of f = (f1; : : : ; fr) with respect to the
homotheties in GL(m) _ Gr
m. This gives the homogeneity conditions
kf1(X1; : : : ;Xs; : : : ;Xr) = f1(kX1; : : : ; ksXs; : : : ; krXr)
...
ksfs(X1; : : : ;Xs; : : : ;Xr) = fs(kX1; (2) : : : ; ksXs; : : : ; krXr)
...
krfr(X1; : : : ;Xs; : : : ;Xr) = fr(kX1; : : : ; ksXs; : : : ; krXr):
Taking into account the homotheties in GL(n), we further _nd
(3)
kf1(X1; : : : ;Xr) = f1(kX1; : : : ; kXr)
...
kfr(X1; : : : ;Xr) = fr(kX1; : : : ; kXr):
Applying the homogeneous function theorem to both (2) and (3), we deduce that
fs is linear in Xs and independent of the remaining coordinates, s = 1; : : : ; r.
Consider furthemore the equivariance with respect to the subgroup GL(m) _
GL(n). This yields that fs corresponds to an equivariant map of Rn SsRm_
into itself. By the generalized invariant tensor theorem, it holds fs = csXs with
any cs 2 R.
For r = 1 we have deduced fp
i = c1Xp
i . For r = 2 consider the equivariance
with respect to the kernel of the jet projection G2
m
_ G2
n
! G1
m
_ G1
n. Taking
into account the coordinate form of the jet composition, we _nd that the action
of an element ((_ij
; ~ai
jk); (_p
q ; bpq
r)) on (Xp
i ;Xp
ij) is _X p
i = Xp
i and
(4) _X p
ij = Xp
ij + bpq
rXq
i Xr
j + Xp
k ~ak
ij :
Then the equivariance condition for fp
ij reads
(5) c2Xp
ij + (c1)2bpq
rXq
i Xr
j + c1Xp
k ~ak
ij = c2(Xp
ij + bpq
rXq
i Xr
j + Xp
k ~ak
ij):
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
32. Jet functors 261
This implies c1 = c2 = 0 or c1 = c2 = 1. Assume by induction that our assertion
holds in the order r 1. Consider the equivariance with respect to the kernel
of the jet projection Gr
m
_ Gr
n
! Gr1
m
_ Gr1
n . The action of an element
((_ij
; 0; : : : ; 0; ~ai
j1:::jr ); (_p
q ; 0; : : : ; 0; bpq
1:::qr )) leaves X1; : : : ;Xr1 unchanged and
it holds
(6) _X p
i1:::ir
= Xp
i1:::ir
+ bpq
1:::qrXq1
i1 : : :Xqr
ir
+ Xp
j ~aj
i1:::ir
:
Then the equivariance condition for fp
i1:::ir
requires
(7) crXp
i1:::ir
+ (c1)rbpq
1:::qrXq1
i1 : : :Xqr
ir
+ c1Xp
j ~aj
i1:::ir
= cr(Xp
i1:::ir
+ bpq
1:::qrXq1
i1 : : :Xqr
ir
+ Xp
j ~aj
i1:::ir
):
This implies cr = c1 = 0 or 1.
For r = 1 we have a homothety fn : X 7! knX, kn 2 R, on each subcategory
Mfm _Mfn _ Mfm _Mf. If we take the value of the transformation
(f1; : : : ; fn; : : : ) on the product of idRm with the injection ia;b : Ra ! Rb,
(x1; : : : ; xa) 7! (x1; : : : ; xa; 0; : : : ; 0), a < b, and apply it to 1-jet at 0 of the map
x1 = t1, x2 = 0; : : : ; xa = 0, (t1; : : : ; tm) 2 Rm, we _nd ka = kb. For r _ 2 we
have on each subcategory either the identity or the contraction. Applying the
latter idea once again, we deduce that the same alternative must take place in
all cases. _
32.2. The construction of the r-th jet prolongation JrY of a _bered manifold
Y ! X can be considered as a bundle functor on the category FMm. This
functor is also denoted by Jr. However, in order to distinguish from 32.1, we
shall use Jr
_b for Jr in the _bered case here.
Proposition. The only natural transformation Jr
_b
! Jr
_b is the identity.
Proof. The construction of product _bered manifolds de_nes an injection Mfm
_Mf ! FMm and the restriction of Jr
_b to Mfm _Mf is Jr. For r = 1,
proposition 31.1 gives a one-parameter family
(1) (yp
i ) 7! (cyp
i )
of possible candidates for the natural transformation J1
_b
! J1
_b. But the transformation
law of yp
i with respect to the kernel of the standard homomorphism
G1
m;n
! G1
m
_ G1
n is _yp
i = yp
i + ap
i . The equivariance condition for (1) reads
ap
i = cap
i , which implies c = 1.
For r _ 2, proposition 32.1 o_ers the contraction and the identity. But the
contraction is clearly not natural on the whole category FMm, so that only the
identity remains. _
32.3. Natural transformations J1J1 ! J1J1. It is well known that the
canonical involution of the second tangent bundle plays a signi_cant role in applications.
A remarkable feature of the canonical involution on TTM is that it
exchanges both the projections pTM : TTM ! TM and TpM : TTM ! TM.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
262 Chapter VII. Further applications
Nowadays, in several problems of the _eld theory the role of the tangent bundle
of a smooth manifold is replaced by the _rst jet prolongation J1Y of a
_bered manifold p: Y ! M. On the second iterated jet prolongation J1J1Y =
J1(J1Y ! M) there are two analogous projections to J1Y , namely the target
jet projection _1 : J1J1Y ! J1Y and the prolongation J1_ : J1J1Y ! J1Y of
the target jet projection _ : J1Y ! Y . Hence one can ask whether there exists
a natural transformation of J1J1Y into itself exchanging the projections _1 and
J1_, provided J1J1 is considered as a functor on FMm;n. But the answer is
negative.
Proposition. The only natural transformation J1J1 ! J1J1 is the identity.
This assertion follows directly from proposition 32.6 below, so that we shall
not prove it separately. It is remarkable that we have a di_erent situation on
the subspace _ J2Y = fX 2 J1J1Y; _1X = J1_(X)g, which is called the second
semiholonomic prolongation of Y . There is a one-parametric family of natural
transformations _ J2 ! _ J2, see 32.5.
32.4. An exchange map. However, one can construct a suitable exchange
map e_ : J1J1Y ! J1J1Y by means of a linear connection _ on the base manifold
M as follows. Interpreting _ as a principal connection on the _rst order
frame bundle P1M of M, we _rst explain how _ induces a map h_ : J1J1Y _
QP1M ! T1m
(T1m
Y ). Every X 2 J1J1Y is of the form X = j1
x_(z), where _ is
a local section of J1Y ! M, and for every u 2 P1
xM we have _(u) = j1
x_(z),
where _ is a local section of P1M _ J1
0 (Rm;M). Taking into account the canonical
inclusion J1Y _ J1(M; Y ), the jet composition _(z) _ _(z) de_nes a local
map M ! J1
0 (Rm; Y ) = T1m
Y , the 1-jet of which j1
x(_(z) _ _(z)) 2 J1
x(M; T1m
Y )
depends on X and _(u) only. Since u 2 J1
0 (Rm;M), we have h_(X; u) =
j1
x(_(z) _ _(z))
_
_ u 2 T1m
T1m
Y . Furthermore, there is a canonical exchange map
_: T1m
T1m
Y ! T1m
T1m
Y , the de_nition of which will be presented in the framework
of the theory of Weil bundles in 35.18. Using _ and h_, we construct a
map e_ : J1J1Y ! J1J1Y .
Lemma. For every X 2 (J1J1Y )y there exists a unique element e_(X) 2
J1J1Y satisfying
(1) _(h_(X; u)) = h~_ (e_(X); u)
for any frame u 2 P1
xM, x = p(y), provided ~_ means the conjugate connection
of _.
Proof consists in direct evaluation, for which the reader is referred to [Kol_a_r,
Modugno, 91]. The coordinate form of e_ is
(2) yp
i = Y p
i ; Y p
i = yp
i ; yp
ij = yp
ji + (yp
k
Y p
k )_kj
i
where Y p
i = @yp=@xi, yp
ij = @yp
i =@xj are the additional coordinates on J1(J1Y
! M).
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
32. Jet functors 263
32.5. Remark. The subbundle _ J2Y _ J1J1Y is characterized by yp
i = Y p
i .
Formula 32.4.(2) shows that the restriction of e_ to _ J2Y does not depend on _,
so that we have a natural map e: _ J2Y ! _ J2Y . Since _ J2Y ! J1Y is an a_ne
bundle, e generates a one-parameter family of natural transformations _ J2 ! _ J2
X 7! kX + (1 k)e(X); k 2 R:
One proves easily that this family represents all natural transformations _ J2 !
_ J2, see [Kol_a_r, Modugno, 91].
32.6. The map e_ was introduced by M. Modugno by another construction, in
which the naturality ideas were partially used. Hence it is interesting to study
the whole problem purely from the naturality point of view.
Our goal is to _nd all natural transformations J1J1Y _ QP1M ! J1J1Y .
Since J1Y ! Y is an a_ne bundle with associated vector bundle V Y T_M,
we can de_ne a map
(1) _ : J1J1Y ! V Y T_M; A 7! _1(A) J1_(A):
On the other hand, proposition 25.2 implies directly that all natural operators
N : QP1M TM T_M T_M form the 3-parameter family
(2) N : _ 7! k1S + k2I ^ S + k3 ^ S I
where S is the torsion tensor of _, ^ S is the contracted torsion tensor and I is
the identity of TM. Using the contraction with respect to TM, we construct a
3-parameter family of maps
(3) h_;N(_)i : J1J1Y ! V Y T_M T_M:
The well known exact sequence of vector bundles over J1Y
(4) 0 ! V Y T_M ! V J1Y
V _
! V Y ! 0
shows that V Y T_MT_M can be considered as a subbundle in V J1Y T_M,
which is the vector bundle associated with the a_ne bundle _1 : J1J1Y ! J1Y .
Proposition. All natural transformations f : J1J1Y ! J1J1Y depending on
a linear connection _ on the base manifold form the two 3-parameter families
(5) I: f = id + h_;N(_)i; II: f = e_ + h_;N(_)i:
Proof. The standard _bers V = (yp
i ; Y p
i ; yp
ij) and Z = (_i
jk) are G2
m;n-spaces
and we have to _nd all G2
m;n-equivariant maps f : V _ Z ! V . The action of
G2
m;n on V is
_yp
i = apq
yq
j ~aj
i + ap
j ~aj
i ; _ Y p
i = apq
Y q
j ~aj
i + ap
j ~aj
i
_yp
ij = apq
yq
kl~aki
~al
j + apq
ryq
kY r
l ~aki
~al
j + ap
qkY q
l ~aki
~al
(6) j+
+ ap
qlyq
k~aki
~al
j + apq
yq
k~ak
ij + ap
kl~aki
~al
j + ap
k~ak
ij
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
264 Chapter VII. Further applications
while the action of G2
m;n on Z is given by 25.2.(3).
The coordinate form of an arbitrary map f : V _ Z ! V is
y = F(y; Y; y2; _)
(7) Y = G(y; Y; y2;_)
y2 = H(y; Y; y2; _)
where y = (yp
i ), Y = (Y p
i ), y2 = (yp
ij ), _ = (_i
jk). Considering equivariance of
(7) with respect to the base homotheties we _nd
kF(y; Y; y2; _) = F(ky; kY; k2y2; k_)
(8) kG(y; Y; y2;_) = G(ky; kY; k2y2; k_)
k2H(y; Y; y2; _) = H(ky; kY; k2y2; k_):
By the homogeneous function theorem, F and G are linear in y, Y , _ and
independent of y2, while H is linear in y2 and bilinear in y, Y , _. The _ber
homotheties then yield
kF(y; Y; _) = F(ky; kY; _)
(9) kG(y; Y;_) = G(ky; kY;_)
kH(y; Y; _) = H(ky; kY; ky2; _):
Comparing (9) with (8) we _nd that F and G are independent of _ and H is
linear in y2 and bilinear in (y; _) and in (Y; _).
Since f is GL(m)_GL(n)-equivariant, we can apply the generalized invariant
tensor theorem. This yields
(10)
Fp
i = ayp
i + bY p
i
Gp
i = cyp
i + dY p
i
Hp
ij = eyp
ij + fyp
ji+
gyp
i _kj
k + hyp
i _kk
j + iyp
j_k
ik + jyp
j_kk
i + kyp
k_k
ij + lyp
k_kj
i+
mY p
i _kj
k + nY p
i _kk
j + pY p
j _k
ik + qY p
j _kk
i + rY p
k _k
ij + sY p
k _kj
i:
The last step consists in expressing the equivariance of (10) with respect to the
subgroup of G2
m;n characterized by ai
j = _ij
, apq
= _p
q . This leads to certain simple
algebraic identities, which are equivalent to (5). _
32.7. Remark. The only map in 32.6.(5) independent of _ is the identity.
This proves proposition 32.3.
If we consider a linear symmetric connection _, then the whole family N(_)
vanishes identically. This implies
Corollary. The only two natural transformations J1J1Y ! J1J1Y depending
on a linear symmetric connection _ on the base manifold are the identity and
e_.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 265
32.8. Remark. The functors _ J2 and J1J1 restricted to the category Mfm _
Mf _ FMm de_ne the so called semiholonomic and non-holonomic 2-jets in the
sense of [Ehresmann, 54]. We remark that all natural transformations of each of
those restricted functors into itself are determined in [Kol_a_r, Vosmansk_a, 87].
Further we remark that [Kurek, to appear b] described all natural transformations
Tr_ ! Ts_ between any two one-dimensional covelocities functors from
12.8. He also determined all natural tensors of type
1
1
_
on Tr_M, [Kurek, to
appear c].
33. Topics from Riemannian geometry
33.1. Our aim is to outline the application of our general procedures to the
study of geometric operations on Riemannian manifolds. Since the Riemannian
metrics are sections of a natural bundle (a subbundle in S2T_), we can always
add the metrics to the arguments of the operation in question instead of specializing
our general approach to categories over manifolds for the category of
Riemannian manifolds and local isometries. In this way, we reduce the problem
to the study of some equivariant maps between the standard _bers, in spite of
the fact that the Riemannian manifolds are not locally homogeneous in the sense
of 18.4. However, at some stage we mostly have to _x the values of the metric
entry by restricting ourselves to the invariance with respect to the isometries and
so we need description of all tensors invariant under the action of the orthogonal
group.
Let us write S2+
T_ for the natural bundle of elements of Riemannian metrics.
33.2. O(m)-invariant tensors. An O(m)-invariant tensor is a tensor B 2
pRm qRm_ satisfying aB = B for all a 2 O(m). The canonical scalar
product on Rm de_nes an O(m)-equivariant isomorphism Rm _= Rm_. This
identi_es B with an element from p+qRm_, i.e. with an O(m)-invariant linear
map p+qRm ! R. Let us de_ne a linear map '_ : 2s Rm ! R, by
'_(v1 _ _ _ v2s) = (v_(1); v_(2)):(v_(3); v_(4)) _ _ _ (v_(2k1); v_(2s));
where ( ; ) means the canonical scalar product de_ned on Rm and _ 2 _2s
is a permutation. The maps '_ are called the elementary invariants. The
fundamental result due to [Weyl, 46] is
Theorem. The linear space of all O(m)-invariant linear maps kRm ! R is
spanned by the elementary invariants for k = 2s and is the zero space if k is
odd.
Proof. We present a proof based on the Invariant tensor theorem (see 24.4),
following the lines of [Atiyah, Bott, Patodi, 73]. The idea is to involve explicitly
all metrics gij 2 S2+
Rm_ and then to look for GL(m)-invariant maps. So together
with an O(m)-invariant map ': k Rm ! R we consider the map _': S2+
Rm_ _
kRm ! R, de_ned by _'(Im; x) = '(x) and _'(G; x) = _'((A1)TGA1; Ax)
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
266 Chapter VII. Further applications
for all A 2 GL(m), G 2 S2+
Rm_. By de_nition, _' is GL(m)-invariant. With
the help of the next lemma, we shall be able to extend the map _' to the whole
S2Rm_ _ kRm.
Let us write V = kRm. The map _' induces a map GL(m) _ V ! R,
(A; x) ! _'(AT A; x) = '(Ax) and this map is extended by the same formula to
a polynomial map f : gl(m) _ V ! R, linear in V . So fx(A) = f(A; x) = '(Ax)
is polynomial and O(m)-invariant for all x 2 V , and f(A; x) = _'(AT A; x) if A
invertible.
Lemma. Let h: gl(m) ! R be a polynomial map such that h(BA) = h(A)
for all B 2 O(m). Then there is a polynomial F on the space of all symmetric
matrices such that h(A) = F(ATA).
Proof. In dimension one, we deal with the well known assertion that each even
polynomial, i.e. h(x) = h(x), is a polynomial in x2. However in higher dimensions,
the proof is quite non trivial. We present only the main ideas and refer
the reader to our source, [Atiyah, Bott, Patodi, 73, p. 323], for more details.
First notice that it su_ces to prove the lemma for non singular matrices, for
then the assertion follows by continuity. Next, if ATA = P with P non singular
and if there is a symmetric Q, Q2 = P, then A lies in the O(m)-orbit of Q.
Indeed, Q is also non singular and B = AQ1 satis_es BTB = Q1ATAQ1 =
Im. So it su_ces to restrict ourselves to symmetric matrices.
Hence we want to _nd a polynomial map g satisfying h(Q) = g(Q2) for all
spymmetric matrices. For every symmetric matrix P, there is the square root
P = Q if we extend the _eld of scalars to its algebraic closure. This can be
computed easily if we express P = BTDB with an orthogonal matrix B and
diagonal matrix D, since then
p
P = BT
p
DB and
p
D is the diagonal matrix
with the square roots of the eigen values of P on its diagonal. But we should
express Q as a universal polynomial in the elements pij of the matrix P. Let us
assume that all eigenvalues _i of P are di_erent. Then we can write
Q =
Xm
i=1
p
_i
Y
j6=i
P _j
_i _j
:
Notice that the eigen values _i are given by rational functions of the elements
pij of P. Thus, in order to make this to a polynomial expression, we have _rst to
extend the _eld of complex numbers to the _eld K of rational functions (i.e. the
elements are ratios of polynomials in pij 's). So for matrices with entries from K,
all eigen values depend polynomially on pij 's. We also need their square roots
to express Q, but next we shall prove that after inserting Q =
p
P into h(Q) all
square roots will factor out. For any _xed P, let us consider the splitting _eld
L over K with respect to the roots of the equation det(P _2) = 0. So
p
P is
polynomial over L. As a polynomial map, h extends to gl(m;L) and the next
sublemma shows that it is in fact O(m;L)-invariant.
Sublemma. Let L be any algebraic extension of R and let f : O(m;L) ! L be
a rational function. If f vanishes on O(m;R) then f is zero.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 267
Proof. The Cayley map C : o(m;R) ! O(m;R) is a birational isomorphism of
the orthogonal group with an a_ne space. Hence there are `enough real points'
to make zero all coe_cients of the rational map. For more details see [Atiyah,
Bott, Patodi, 73] _
Now the basic fact is, that for any automorphism _ : L ! L of the Galois
group of L over K we have (_Q)2 = _P = P and since both Q and _Q are symmetric,
B = _QQ1 is orthogonal. Hence we get _h(Q) = h(_Q) = h(BQ) =
h(Q). Since this holds for all _, h(Q) lies in K and so h(Q) = g(Q2) for a
rational function g.
The latter equality remains true if P is a real symmetric matrix such that all
its eigen values are distinct and the denominator of g(P) is non zero. If g = F=G
for two polynomials F and G, we get F(ATA) = h(A)G(ATA). If we choose A
so that G(ATA) = 0, we get F(ATA) = 0. Hence g is a globally de_ned rational
function without poles and so a polynomial.
Thus, we have found a polynomial F on the space of symmetric matrices
such that h(A) = F(ATA) holds for a Zariski open set in gl(m). This proves
our lemma. _
Let us continue in the proof of the Weyl's theorem. By the lemma, every
fx satis_es fx(A) = gx(ATA) for certain polynomial gx and so we get a polynomial
mapping g : S2Rm_ _ V ! R linear in V . For all B;A 2 GL(m;C) we
have g((B1)TATAB1;Bx) = f(AB1;Bx) = f(A; x) = g(AT A; x) and so
g : S2Rm_ _ V ! R is GL(m)-invariant. Then the composition of g with the
symmetrization yields a polynomial GL(m)-invariant map 2Rm__kRm ! R,
linear in the second entry. Each multi homogeneous component of degree s+1 in
the sense of 24.11 is also GL(m)-invariant and so its total polarization is a linear
GL(m)-invariant map H: 2sRm_kRm ! R. Hence, by the Invariant tensor
theorem, k = 2s and H is a sum of complete contractions over possible permutations
of indices. Since the original mapping ' is given by '(x) = g(Im; x),
Weyl's theorem follows. _
33.3. To explain the coordinate form of 33.2, it is useful to consider an arbitrary
metric G = (gij) 2 S2+
Rm_. Let O(G) _ GL(m) be the subgroup
of all linear isomorphisms preserving G, so that O(m) = O(Im). Clearly,
theorem 33.2 holds for O(G)-invariant tensors as well. Every O(G)-invariant
tensor B = (Bi1:::ip
j1:::jp
) 2 pRm qRm_ induces an O(G)-invariant tensor
gi1k1 : : : gipkpBk1:::kp
j1:::jq
2 p+qRm_. Hence theorem 33.2 implies that all O(G)-
invariant tensors in pRm qRm_ with p + q even are linearly generated by
gi1k1 : : : gipkpg_(k1)_(k2) : : : g_(jq1)_(jq)
where gikgjk = _ij
, gij = gji, for all permutations _ of p + q letters.
Consequently, all O(G)-equivariant tensor operations are generated by: tensorizing
by the metric tensor G: Rm ! Rm_ or by its inverse ~G : Rm_ ! Rm,
applying contractions and permutations of indices, and taking linear combinations.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
268 Chapter VII. Further applications
33.4. Our next main goal is to prove the famous Gilkey theorem on natural
exterior forms on Riemannian metrics, i.e. to determine all natural operators
S2+
T_ _pT_. This will be based on 33.2 and on the reduction theorems from
section 28. But since the resulting forms come from the Levi-Civit_a connection
via the Chern-Weil construction, we _rst determine all natural operators transforming
linear symmetric connections into exterior forms. This will help us to
describe easily the metric operators later on.
Let us start with a description of natural tensors depending on symmetric
linear connections, i.e. natural operators Q_P1 T(p;q), where T(p;q)Rm =
Rm _ pRm qRm_. Each covariant derivative of the curvature R() 2
C1(TM T_M _2T_M) of the connection on M is natural. Further every
tensor multiplication of two natural tensors and every contraction on one covariant
and one contravariant entry of a natural tensor give new natural tensors.
Finally we can tensorize any natural tensor with a GL(m)-invariant tensor, we
can permute any number of entries in the tensor products and we can repeat
each of these steps and take linear combinations.
Lemma. All natural operators Q_P1 T(p;q) are obtained by this procedure.
In particular, there are no non zero operators if q p = 1 or q p < 0.
Proof. By 23.5, every such operator has some _nite order r and so it is determined
by a smooth Gr+2
m -equivariant map f : Trm
Q ! V , where Q is the standard
_ber of the connection bundle and V = pRmqRm_. By the proof of the theorem
28.6, there is a G1
m-equivariant map g : Wr1 ! V such that f = g _Cr1.
Here Wr1 = W _ : : : _ Wr1, W = Rm Rm_ _2Rm_, Wi = W iRm_,
i = 1; : : : ; r1. Therefore the coordinate expression of a natural tensor is given
by smooth maps
!i1:::ip
j1:::jq
(Wi
jkl; : : : ;Wi
jklm1:::mr1 ):
Hence we can apply the Homogeneous function theorem (see 24.1). The action
of the homotheties c1_ij
2 G1
m gives
cqp!i1:::ip
j1:::jq
(Wi
jkl; : : : ;Wi
jklm1:::mr1 ) = !i1:::ip
j1:::jq
(c2Wi
jkl; : : : ; cr+1Wi
jklm1:::mr1 ):
Hence the !'s must be sums of homogeneous polynomials of degrees ds in the
variables Wi
jklm1:::ms
satisfying
(1) 2d0 + _ _ _ + (r + 1)dr1 = q p:
Now we can consider the total polarization of each multi homogeneous component
and we obtain linear mappings
Sd0W _ _ _ Sdr1Wr1 ! V:
According to the Invariant tensor theorem, all the polynomials in question are
linearly generated by monomials obtained by multiplying an appropriate number
of variables Wi
jkl_ and applying some of the GL(m)-equivariant operations.
If q = p, then the polynomials would be of degree zero, and so only the
GL(m)-invariant tensors can appear. If q p = 1 or q p < 0, there are no non
negative integers solving (1). _
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 269
33.5. Natural forms depending on linear connections. To determine
the natural operators Q_P1 _qT_, we have to consider the case p = 0 and
apply the alternation to the subscripts. It is well known that the Chern-Weil
construction associates a natural form to every polynomial P which is de_ned
on Rm Rm_ and invariant under the action of GL(m). This natural form is
obtained by substitution of the entries of the matrix valued curvature 2-form
R for the variables and taking the wedge product for multiplication. So if P
is homogeneous of degree j, then P(R) is a natural 2j-form. Let us denote by
!q the form obtained from the tensor product of q copies of the curvature R
by taking its trace and alternating over the remaining entries. In coordinates,
!q = (Rkq
k1abRk1
k2cd : : :Rkq1
kqef ), where we alternate over all indices a; : : : ; f. One
_nds easily that the polynomials Pq depending on the entries of the matrix 2-
form R correspond to the homogeneous components of degree q in det(Im + R)
and so the forms !q equal the Chern forms cq up to the constant factor (i=(2_))q.
The wedge product on the linear space of all natural forms depending on
connections de_nes the structure of a graded algebra.
Theorem. The algebra of all natural operators Q_P1 _m
p=0_pT_ is generated
by the Chern forms cq.
In particular, there are no natural forms with odd degrees and consequently
all natural forms are closed.
Proof. We have to continue our discussion from the proof of the lemma 33.4.
However, we need some relations on the absolute derivatives Rij
klm1:::ms
of the
curvature tensor. First recall the antisymmetry, the _rst and the second Bianchi
identity, cf. 28.5
Rij
kl = Rij
lk (1)
Rij
kl + Rik
lj + Ri
(2) ljk = 0
Rij
klm + Rij
lmk + Rij
mkl (3) = 0
Lemma. The alternation of Rij
klm1:::ms
over any 3 indices among the _rst four
subscripts is zero.
Proof. Since the covariant derivative commutes with the tensor operations like
the permutation of indices, it su_ces to discuss the variables Rij
kl and Rij
klm.
By (2), the alternation over the subscripts in Rij
kl is zero and (3) yields the same
for the alternation over k, l, m in Rij
klm. In view of (1), it remains to discuss
the alternation of Rij
klm over j, l, m. (1) implies Rij
kml = Rij
mkl and so we
can rewrite this alternation as follows
Rij
klm + Rij
mkl + Rij
lmk
Rij
lmk
+Ri
mkjl + Ri
mlkj + Ri
mjlk
Ri
mjlk
+Ri
lkmj + Ri
ljkm + Ri
lmjk
Ri
lmjk:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
270 Chapter VII. Further applications
The _rst three entries on each row form a cyclic permutation and hence give
zero. The same applies to the last column. _
Now it is easy to complete the proof of the theorem. Consider _rst a monomial
containing at least one quantity Rij
klm1:::ms
with s > 0. Then there exists
one term of the product with three free subscripts among the _rst four ones
or one term Rij
kl with all free subscripts, so that the monomial vanishes after
alternation. Further, (1) and (2) imply Rij
kl
Ri
lkj = Rik
lj . Hence we can
restrict ourselves to contractions with the _rst subscripts and so all the possible
natural forms are generated by the expressions Rkq
k1abRk1
k2cd : : :Rkq1
kqef where the
indices a; : : : ; f remain free for alternation. But these are coordinate expressions
of the forms !q. _
33.6. Characteristic classes. The dimension of the homogeneous component
of the algebra of natural forms of degree 2s equals the number _(s) of the partitions
of s into sums of positive integers. Since all natural forms are closed, they
determine cohomology classes in the De Rham cohomologies of the underlying
manifolds. It is well known from the Chern-Weil theory that these classes do
not depend on the connection. This can be deduced as follows.
Consider two linear connections , _
expressed locally by i
j , _i
j
2 (T_M
TM)T_M, and their curvatures Rij
, _R
ij
2 (T_MTM)_2T_M. Write t =
t_
+ (1 t) and analogously Rt for the curvatures. Let Pq be the polynomial
de_ning the form !q and Q be its total polarization. We de_ne _q(; _) =
q
R 1
0 Q(_
;Rt; : : : ;Rt)dt. The structure equation yields d
dtRt = d
dt (dt)
d
dtt ^ t t ^ d
dtt = d(_
) and we calculate easily in normal coordinates
!q(_) !q() =
Z 1
0
d
dt
Q(Rt; : : : ;Rt)dt = q
Z 1
0
Q( d
dt
Rt;Rt; : : : ;Rt)dt
= q
Z 1
0
dQ(_
;Rt; : : : ;Rt)dt = d_q(; _):
In fact, _q is one of many natural operators Q_P1 _ Q_P1 _2q1T_ and the
integration helps us to _nd the proper linear combination of more elementary
operators which are obtained by a procedure similar to that from 33.4{33.5. The
form _q(; _) is called the transgression.
33.7. Natural forms on Riemannian manifolds. Since there is the natural
Levi-Civit_a connection, we can evaluate the natural forms from 33.5 using the
curvature of this connection. In this case 28.14.(3) holds, i.e.
(1) ginWn
jklm1:::mr = gjnWn
iklm1:::mr :
For gij = _ij , r = 0, this implies
(2) Rij
kl = Rj
ikl
and so the contractions in a monomial Rkq
k1abRk1
k2cd : : :Rkq1
kqef yield zero if q is
odd. The natural forms pj = (2_)2j!2j are called the Pontryagin forms. The
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 271
dimension of the homogeneous component of degree 4s of the algebra of forms
generated by the Pontryagin forms is _(s), cf. 33.6.
If we assume the dependence of the natural operators on the metric, then
every two indices of any tensor can be contracted. In particular, the complete
contractions of covariant derivatives of the curvature of the Levi-Civit_a connection
give rise to natural functions of all even orders grater then one. Composing
k natural functions with any _xed smooth function Rk ! R, we get a new natural
function. Since every natural form can be multiplied by any natural function
without loosing naturality, we see that there is no hope to describe all natural
forms in a way similar to 33.5. However, in Riemannian geometry we often meet
operations with a sort of homogeneity with respect to the change of the scale of
the metric and these can be described in more details.
Our operators will have several arguments as a rule and we shall use the
following brief notation in this section: Given several natural bundles Fa; : : : ; Fb,
we write Fa _: : :_Fb for the natural bundle associating to each m-manifold M
the _bered product FaM_M: : :_MFbM and similarly on morphisms. (Actually,
this is the product in the category of functors, cf. 14.11.) Hence D: F1_F2 G
means a natural operator transforming couples of sections from C1(F1M) and
C1(F2M) to sections from C1(GM) (which is also denoted by D: F1_F2 G
in this book). Analogously, given natural operators D1 : F1 G1 and D2 : F2
G2, we use the symbol D1 _ D2 : F1 _ F2 G1 _ G2.
De_nition. Let E and F be natural bundles over m-manifolds. We say that a
natural operator D: S2+
T_ _ E F is conformal, if D(c2g; s) = D(g; s) for all
metrics g, sections s, and all positive c 2 R. If F is a natural vector bundle and
D satis_es D(c2g) = c_D(g), then _ is called the weight of D.
Let us notice that the weight of the metric gij is 2 (we consider the inclusion
g : S2+
T_ S2T_), that of its inverse gij is 2, while the curvature and all its
covariant derivatives are conformal.
33.8. Gilkey theorem. There are no non zero natural forms on Riemannian
manifolds with a positive weight. The algebra of all conformal natural forms on
Riemannian manifolds is generated by the Pontryagin forms.
33.9. Let us start the proof with a discussion on the reduction procedure developed
in section 28. Even if we have no estimate on the order, we can get
an analogous result. Consider an arbitrary natural operator Q_P1 _ E F.
By the non-linear Peetre theorem, D is of order in_nity and so it is determined
by the restriction D of its associated mapping J1((Q_P1 _ E)Rm) ! FRm
to the _ber over the origin. Moreover, we obtain an open _ltration of the
whole _ber J1
0 ((Q_P1 _ E)Rm) consisting of maximal G1
m-invariant open subsets
Uk where the associated mapping D factorizes through Dk : _1
k (Uk) _
Jk
0 ((Q_P1 _ E)Rm) ! F0Rm. Now, we can apply the same procedure as in
the section 28 to this invariant open submanifolds _1
k (Uk).
Let F be a _rst order bundle functor on Mfm, E be an open natural sub
bundle of a vector bundle functor _E on Mfm. The curvature and its covariant
derivatives are natural operators _k : Q_P1 Rk, with values in tensor bundles
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
272 Chapter VII. Further applications
Rk, RkRm = Rm_Wk, W0 = RmRm__2Rm_, Wk+1 = WkRm_. Similarly,
the covariant di_erentiation of sections of E forms natural operators dk : Q_P1_
E Ek, where E0 = _E , E0Rm =: Rm_V0, d0 is the inclusion, EkRm = Rm_Vk,
Vk+1 = Vk Rm_. Let us write Dk = (_0; : : : ; _k2; d0; : : : ; dk) : Q_P1 _ E
Rk2 _ Ek, where Rl = R0 _ : : : _ Rl, El = E0 _ : : : _ El. All Dk are natural
operators. In 28.8 we de_ned the Ricci sub bundles Zk _ Rk2 _ Ek and we
know Dk : Q_P1 _ E Zk.
Let us further de_ne the functor Z1 as the inverse limit of Zk, k 2 N, with
respect to the obvious natural transformations (projections) pk`
: Zk ! Z`, k > `,
and similarly D1: Q_P1 _E Z1. As a corollary of 28.11 and the non linear
Peetre theorem we get
Proposition. For every natural operator D: Q_P1 _ E F there is a unique
natural transformation ~D : Z1 ! F such that D = ~D_D1. Furthermore, for every
m-dimensional compact manifold M and every section s 2 C1(Q_P1M _M
EM), there is a _nite order k and a neighborhood V of s in the Ck-topology
such that ~DMj(D1)M(V ) = (_1
k )_(~Dk)M, for some (~Dk)M : (Dk)M(V ) !
C1(ZkM), and DMjV = (~Dk)M _ (Dk)MjV .
In words, a natural operator D: Q_ _ E F is determined in all coordinate
charts of an arbitrary m-dimensional manifoldM by a universal smooth mapping
de_ned on the curvatures and all their covariant derivatives and on the sections
of EM and all their covariant derivatives, which depends `locally' only on _nite
number of these arguments.
33.10. The Riemannian case. In section 28, we also applied the reduction
procedure to operators depending on Riemannian metrics and general vector
_elds. In fact we have viewed the operators D: S2+
T_ _ E F as operators
_D
: Q_P1 _ (S2+
T_ _ E) F independent of the _rst argument and we have
used the Levi-Civit_a connection : S2+
T_ Q_P1 to write D as a composition
D = _D _ (; id). Since the covariant derivatives of the metric with respect to
the metric connection are zero, we can restrict ourselves to sub bundles in the
Ricci subspaces corresponding to the bundle S2+
T_ _ E, which are of the form
S2+
T_ _ Zk with Zk _ Rk2 _ Ek, cf. 28.14. Let us notice that the bundles
ZkM involve the curvature of the Riemannian connection on M, its covariant
derivatives, and the covariant derivatives of the sections of EM. Similarly as
above, we de_ne the inverse limits Z1 and D1 and as a corollary of the non
linear Peetre theorem and 28.15 we get
Corollary. For every natural operator D: S2+
T_ _ E F there is a natural
transformation ~D : S2+
T_ _ Z1 ! F such that D = ~D _ D1 _ (; id).
Furthermore, for every m-dimensional compact manifold M and every section
s 2 C1(S2+
T_M _M EM), there is a _nite order k and a neighborhood V of s
in the Ck-topology such that ~DMj(D1 _ (; id))M(V ) = (_1
k )_(~Dk)M, where
(~Dk)M : (Dk _ (; id))M(V ) ! C1(ZkM), and DMjV = (~Dk)M _ (Dk)M _
(; id)MjV .
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 273
33.11. Polynomiality. Since the standard _ber V0 of E0 is embedded identically
into Zk
0Rm by the associated map to the operator Dk, we can use 28.16 and
add the following proposition to the statements of 33.9, or 33.10, respectively.
Corollary. The operator D is polynomial if and only if the operators ~Dk are
polynomial. Further D is polynomial with smooth real functions on the values of
E0, or S2+
T_, as coe_cients if and only if the operators ~D k are polynomial with
smooth real functions on the values of E0, or S2+
T_, as coe_cients, respectively.
33.12. Natural operators D: S2+
T_ T(p;q). According to 33.9 we _nd G1
m-
invariant open subsets Uk in J1
0 (S2+
T_Rm) forming a _ltration of the whole jet
space, such that on these subsets D factorizes through smooth Gk+1
m -equivariant
mappings
fi1:::ip
j1:::jq
= fi1:::ip
j1:::jq
(gij ; : : : ; gij`1:::`k )
de_ned on _1
k Uk. For large k's, the action of the homotheties c1_ij
on g's is
well de_ned and we get
(1) cqpfi1:::ip
j1:::jq
(gij ; : : : ; gij`1:::`k ) = fi1:::ip
j1:::jq
(c2gij ; : : : ; c2+kgij`1:::`k ):
Now, let us add the assumption that D is homogeneous with weight _, choose
the change g 7! c2g of the scale of the metric and insert this new metric into
(1). We get
cqp��_fi1:::ip
j1:::jq
(gij ; : : : ; gij`1:::`k ) = fi1:::ip
j1:::jq
(gij ; c1gij;`1 ; : : : ; ckgij`1:::`k ):
This formula shows that the mappings fi1:::ip
j1:::jq
are polynomials in all variables
except gij with functions in gij as coe_cients.
According to 33.11 and 28.16, the map D is on Uk determined by a polynomial
mapping
! = (!i1:::ip
j1:::jq
(gij ;Wi
jkl; : : : ;Wi
jklm1:::mk2 ))
which is G1
m-equivariant on the values of the covariant derivatives of the curvatures
and the sections. If we apply once more the equivariance with respect to
the homothety x 7! c1x and at the same time the change of the scale of the
metric g 7! c2g, we get
cqp_!i1:::ip
j1:::jq
(gij ;Rij
kl; : : : ;Rij
klm1:::mk2 ) =
= !i1:::ip
j1:::jq
(gij ; c2Rij
kl; : : : ; ckRij
klm1:::mk2 ):
This homogeneity shows that the polynomial functions !i1:::ip
j1:::jq
must be sums of
homogeneous polynomials with degrees a` in the variables Rij
klm1:::m`
satisfying
(2) 2a0 + _ _ _ + kak2 = q p _
and their coe_cients are functions depending on gij 's.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
274 Chapter VII. Further applications
Now, we shall _x gij = _ij and use the O(m)-equivariance of the homogeneous
components of the polynomial mapping !. For this reason, we shall switch to
the variables Rijklm1:::ms = giaRa
jklm1:::ms
. Using the standard polarization technique
and H. Weyl's theorem, we get that each multi homogeneous component
in question results from multiplication of variables Rijklm1;::: ;ms , s = 0; 1; : : : ; r,
and application of some O(m)-equivariant tensor operations on the target space.
Hence our operators result from a _nite number of the following steps.
(a) take tensor product of arbitrary covariant derivatives of the curvature
tensor
(b) tensorize by the metric or by its inverse
(c) apply arbitrary GL(m)-equivariant operation
(d) take linear combinations.
33.13. Remark. If qp = _+1, then there is no non negative integer solution
of 33.12.(2) and so all natural tensors in question are zero. The case q = 2,
p = 1, _ = 0 implies that the Levi-Civit_a connection is the only conformal
natural connection on Riemannian manifolds.
Indeed, the di_erence of two such connections is a natural tensor twice covariant
and once contravariant, and so zero.
33.14. Consider now _pRm_ as the target tensor space. So in the above procedure,
all indices which were not contracted must be alternated at the end. Since
the metric is a symmetric tensor, we get zero whenever using the above step
(b) and alternating over both indices. But contracting over any of them has no
proper e_ect, for _ijRjklnm1;::: ;ms = Riklnm1;::: ;ms . So we can omit the step (b)
at all.
The _rst Bianchi identity and 33.7.(1) imply Rijkl = Rklij . Then the lemma
in 33.5 and 33.7.(1) yield
Lemma. The alternation of Rijklm1:::ms , 0 _ s, over arbitrary 3 indices among
the _rst four or _ve ones is zero.
Consider a monomial P in the variables Rijkl_ with degrees as in Rijklm1:::ms .
In view of the above lemma, if P remains non zero after all alternations, then we
must contract over at least two indices in each Rijkl_ and so we can alternate
over at most 2a0+_ _ _+kak2 indices. This means p _ 2a0+_ _ _+kak2 = p_.
Consequently _ _ 0 if there is a non zero natural form with weight _. This proves
the _rst assertion of theorem 33.8.
Let _ = 0. Since the weight of gij is 2, any contraction on two indices
in the monomial decreases the weight of the operator by 2. Every covariant
derivative Rijklm1:::ms of the curvature has weight 2. So we must contract on
exactly two indices in each Rijklm1:::ms which implies there are s + 2 of them
under alternation. But then there must appear three alternated indices among
the _rst _ve if s 6= 0. This proves a1 = _ _ _ = ak2 = 0, so that p = 2a0. Hence
all the natural forms have even degrees and they are generated by the forms
!q, cf. 33.5. As we deduced in 33.7, these forms are zero if their degree is not
divisible by four.
This completes the proof of the theorem 33.8. _
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33. Topics from Riemannian geometry 275
33.15. Remark. The original proof of the Gilkey theorem assumes a polynomial
dependence of the natural forms on a _nite number of the derivatives
gij;_ of the metric and on the entries of the inverse matrix gij , but also the
homogeneity in the weight, [Gilkey, 73]. Under such polynomiality assumption,
our methods apply to all natural tensors. In particular, it follows easily that
the Levi-Civit_a connection is the only second order polynomial connection on
Riemannian manifolds. Of course, the latter is not true in higher orders, for we
can contract appropriate covariant derivatives of the curvature and so we get
natural tensors in T T_ T_ of orders higher than two.
33.16. Operations on exterior forms. The approach from 33.4{33.5 can be
easily extended to the study of all natural operators D: Q_P1 _ T(s;r) T(q;p)
with s < r or s = r = 0. This was done in [Slov_ak, 92a], we shall present only
the _nal results. If we omit the assumption on s and r, we have to assume the
polynomiality.
Theorem. All natural operators D: Q_P1_T(s;r) T(q;p), s < r, are obtained
by a _nite iteration of the following steps: take tensor product of arbitrary
covariant derivatives of the curvature tensor or the covariant derivatives of the
tensor _elds from the domain, apply arbitrary GL(m)-equivariant operation,
take linear combinations. In the case s = r = 0 we have to add one more
step, the compositions of the functions from the domain with arbitrary smooth
functions of one real variable.
The algebra of all natural operators D: Q_P1 _ T(0;r) _T_, r > 0, is
generated by the alternation, the exterior derivative d and the Chern forms cq.
The algebra of all natural operators D: Q_P1 _ T(0;0) _T_ is generated
by the compositions with arbitrary smooth functions of one real variable, the
exterior derivative d and the Chern forms cq.
The proof of these results follows the lines of 33.4{33.5 using two more lemmas:
First, the alternation on all indices of the second covariant derivative r2t of an
arbitrary tensor t 2 C1(sRm_) is zero (which is proved easily using the Bianchi
and Ricci identities) and , second, the alternation of the _rst covariant derivative
of an arbitrary tensor t 2 C1(sRm_) coincides with the exterior di_erential of
the alternation of t (this well known fact is proved easily in normal coordinates).
33.17. Operations on exterior forms on Riemannian manifolds. A modi
_cation of our proof of the Gilkey theorem for operations on exterior forms on
Riemannian manifolds, which is also based on the two lemmas mentioned above,
appeared in [Slov_ak, 92a]. The equality 33.7.(2) on the Riemannian curvatures
can be expressed as Rijkl = Rjikl, and this holds for curvatures of metrics
with arbitrary signatures. This observation extends our considerations to pseudoriemannian
manifolds, see [Slov_ak, 92b]. In particular, our proof of the Gilkey
theorem extends to the classi_cation of natural forms on pseudoriemannian manifolds.
Let us write S2
regT_ for the bundle functor of all non degenerate symmetric
two-forms. The de_nition of the weight of the operators depending on metrics
and the de_nition of the Pontryagin forms extend obviously to the pseudoriemannian
case. All the considerations go also through for metrics with any _xed
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
276 Chapter VII. Further applications
signature.
Theorem. All natural operators D: S2
regT_ _ T(s;r) T(q;p), s < r, homogeneous
in weight are obtained by a _nite iteration of the following steps: take
tensor product of arbitrary covariant derivatives of the curvature tensor or the
covariant derivatives of the tensor _elds from the domain, tensorize by the metric
or its inverse, apply arbitrary GL(m)-equivariant operation, take linear combinations.
In the case s = r = 0 we have to add one more step, the compositions
of the functions from the domain with arbitrary smooth functions of one real
variable.
There are no non zero operators D: S2
regT_ _ T(0;r) _T_, r _ 0, with a
positive weight. The algebra of all conformal natural operators S2
regT__T(0;r)
_T_, r > 0, is generated by the alternation, the exterior derivative d and the
Pontryagin forms pq.
The algebra of all conformal natural operators D: S2
regT_ _ T(0;0) _T_
is generated by the compositions with arbitrary smooth functions of one real
variable, the exterior derivative d and the Pontryagin forms pq.
The discussion from the proof of these results can be continued for every _xed
negative weight. In particular, the situation is interesting for _ = 2 and linear
operators D: _pT_ _pT_ depending on the metric. Beside the compositions
d _ _ and _ _ d of the exterior di_erential d and the well known codi_erential
_ : _p _p1 (the Laplace-Beltrami operator is _ = _ _ d + d _ _), there are
only three other generators: the multiplication by the scalar curvature, the contraction
with the Ricci curvature and the contraction with the full Riemmanian
curvature. This classi_cation was derived under some additional assumptions in
[Stredder, 75], see also [Slov_ak, 92b].
33.18. Oriented pseudoriemannian manifolds. It is also quite important
in Riemannian geometry to know what are the operators natural with respect to
the orientation preserving local isometries. We shall not go into details here since
this would require to extend the description from 33.2 to all SO(m)-invariant
linear maps and then to repeat some steps of the proof of the Gilkey theorem.
This was done in [Stredder, 75] (for the polynomial forms and Riemannian
manifolds), and in [Slov_ak, 92b]. Let us only remark that on oriented pseudoriemannian
manifolds we have a natural volume form !: S2
regT_ _mT_ and
natural transformations _: _pT_ ! _mpT_. All natural operators on oriented
pseudoriemannian manifolds homogeneous in the weight are generated by those
described above, the volume form !, and the natural transformations _.
As an example, let us draw a diagram which involves all linear natural conformal
operators on exterior forms on oriented pseudoriemannian manifolds of
even dimensions which do not vanish on at pseudoriemannian manifolds, up to
the possible omitting of the d's on the sides in the operators indicated by the
long arrows. (More explicitely, we do not consider any contribution from the
curvatures.) The symbols p refer, as usual, to the p-forms, the plus and minus
subscripts indicate the splitting into the selfdual and anti-selfdual forms in the
degree 1
2m.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 277
In the even dimensional case, there are no natural conformal operators between
exterior forms beside the exterior derivatives. For the proofs see [Slov_ak,
92b].
We should also remark that the name `conformal' is rather misleading in the
context of the natural operators on conformal (pseudo-) Riemannian manifolds
since we require the invariance only with respect to constant rescaling of the
metric (cf. the end of the next section). On the other hand, each natural operator
on the conformal manifolds must be conformal in our sense.
p
+h
hj
d '')
d+
0 w d 1 w d _ _ _ w d p1 p+1 w d _ _ _ w d m1 w d m
p
AAAC
d
_
d
Dp1=d_d=d_d+d_d
u
D1=d_(_d)m3
u
D0=d_(_d)m1
u
33.19. First order operators. The whole situation becomes much easier if
we look for _rst order natural operators D: S2+
T_ (F;G), where F and G are
arbitrary natural bundles, say of order r. Namely, every metric g on a manifold
M satis_es gij = _ij and @gij
@xk = 0 at the center of any normal coordinate chart.
Therefore, if D, _D are two such operators and if their values DRm(g), _DRm(g)
on the canonical Euclidean metric g on Rm coincide on the _ber over the origin,
then D = _D. Hence the whole classi_cation problem reduces to _nding maps
between the standard _bers which are equivariant with respect to the action of
the subgroup O(m) o Br
1
_ G1
m o Br
1 = Gr
m. In fact we used this procedure in
section 29.
Let us notice that the natural operators on oriented Riemannian manifolds
are classi_ed on replacing O(m)oBr
1 by SO(m)oBr
1. If we modify 29.7 in such
a way, we obtain (cf. [Slov_ak, 89])
Proposition. All _rst order natural connections on oriented Riemannian manifolds
are
(1) The Levi-Civit_a connection , if m > 3 or m = 2
(2) The one parametric family + kD1 where D1 means the scalar product
and k 2 R, if m = 1
(3) The one parametric family +kD3 where D3 means the vector product
and k 2 R, if m = 3.
33.20. Natural metrics on the tangent spaces of Riemannian manifolds.
At the end of this section, we shall describe all _rst order natural operators
transforming metrics into metrics on the tangent bundles. The results were
proved in [Kowalski, Sekizava, 88] by the method of di_erential equations. Let
us start with some notation.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
278 Chapter VII. Further applications
We write _M : TM ! M for the natural projection and F for the natural
bundle with FM = __
M(T_T_)M ! M, Ff(Xx; gx) = (Tf:Xx; (T_T_)f:gx)
for all manifolds M, local di_eomorphisms f, Xx 2 TxM, gx 2 (T_ T_)xM.
The sections of the canonical projection FM ! TM are called the F-metrics
in literature. So the F-metrics are mappings TM _ TM _ TM ! R which are
linear in the second and the third summand. We _rst show that it su_ces to
describe all natural F-metrics, i.e. natural operators S2+
T_ (T; F).
There is the natural Levi-Civit_a connection : TM _ TM ! TTM and the
natural equivalence _ : TM_TM ! V TM. There are three F-metrics, naturally
derived from sections G: TM ! (S2T_)TM. Given such G on TM, we de_ne
(1)
1(G)(u;X; Y ) = G((u;X); (u; Y ))
2(G)(u;X; Y ) = G((u;X); _(u; Y ))
3(G)(u;X; Y ) = G(_(u;X); _(u; Y )):
Since G is symmetric, we know also G(_(u;X); (u; Y )) = 2(G)(u; Y;X). Notice
also that 1 and 3 are symmetric.
The connection de_nes the splitting of the second tangent space into the
vertical and horizontal subspaces. We shall write Xx;u = Xh
u + Xv
u for each
Xx;u 2 TuTM, _(u) = x. Since for every Xx;u there are unique vectors Xh 2
TxM, Xv 2 TxM such that (u;Xh) = Xh
u and _(u;Xv) = Xv
u, we can recover
the values of G from the three F-metrics i,
G(Xx;u; Yx;u) = 1(G)((2) u;Xh; Y h) + 2(G)(u;Xh; Y v)
+ 2(G)(u; Y h;Xv) + 3(G)(u;Xv; Y v):
Lemma. The formulas (1) and (2) de_ne a bijection between triples of natural
F-metrics where the _rst and the third ones are symmetric, and the natural
operators S2+
T_ (S2T_)T. _
33.21. Let us call every section G: TM ! (S2T_)TM a (possibly degenerated)
metric. If we _x an F-metric _, then there are three distinguished constructions
of a metric G.
(1) If _ symmetric, we choose 1 = 3 = _, 2 = 0. So we require that G
coincides with _ on both vertical and horizontal vectors. This is called
the Sasaki lift and we write G = _s. If _ is non degenerate and positive
de_nite, the same holds for _s.
(2) We require that G coincides with _ on the horizontal vectors, i.e. we put
1 = _, 2 = 3 = 0. This is called the vertical lift and G is a degenerate
metric which does not depend on the underlying Riemannian metric. We
write G = _v.
(3) The horizontal lift is de_ned by 2 = _, 1 = 3 = 0 and is denoted by
G = _h. If _ positive de_nite, then the signature of G is (m;m).
We can reformulate the lemma 33.20 as
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
33. Topics from Riemannian geometry 279
Proposition. There is a bijective correspondence between the triples of natural
F-metrics (_; _; ), where _ and are symmetric, and natural (possibly
degenerated) metrics G on the tangent bundles given by
G = _s + _h + v: _
33.22. Proposition. All _rst order natural F-metrics _ in dimensions m > 1
form a family parameterized by two arbitrary smooth functions _, _ : (0;1) ! R
in the following way. For every Riemannian manifold (M; g) and tangent vectors
u, X, Y 2 TxM
(1) _(M;g)(u)(X; Y ) = _(g(u; u))g(X; Y ) + _(g(u; u))g(u;X)g(u; Y ):
If m = 1, then the same assertion holds, but we can always choose _ = 0.
In particular, all _rst order natural F-metrics are symmetric.
Proof. We have to discuss all O(m)-equivariant maps _: Rm ! Rm_ Rm_.
Denote by g0 =
P
i dxi dxi the canonical Euclidean metric and by j j the
induced norm. Each vector v 2 Rm can be transformed into jvj @
@x1
__
0. Hence _
is determined by its values on the one-dimensional subspace spanned by @
@x1
__
0.
Moreover, we can also change the orientation on the _rst axis, i.e. we have to
de_ne _ only on t @
@x1
__
0 with positive reals t.
Let us consider the group G of all linear orthogonal transformations keeping
@
@x1
__
0 _xed. So for every t 2 R the tensor _(t) = _(t @
@x1 ) 2 Rm_ Rm_ is
G-invariant. On the other hand, every such smooth map _ determines a natural
F-metric.
So let us assume sijdxi dxj is G-invariant. Since we can change the orientation
of any coordinate axis except the _rst one, all sij with di_erent indices
must be zero. Further we can exchange any couple of coordinate axis di_erent
from the _rst one and so all coe_cients at dxidxi, i 6= 1, must coincide. Hence
all G-invariant tensors are of the form
(2) _dx1 dx1 + _g0:
The reals _ and _ are independent, if m > 1. In dimension one, G is the trivial
group and so the whole one dimensional tensor space consists of G-invariant
tensors.
Thus, our mapping _ is de_ned by (2) with two arbitrary smooth functions
_ and _ (and they can be reduced to one if m = 1). Given v = t @
@x1
__
0, we can
write
_(Rm;g0)(v)(X; Y ) = _(jvj)(X; Y ) = _(jvj)g0(X; Y )+_(jvj)jvj2g0(v;X)g0(v; Y )
In order to prove that all natural F-metrics are of the form (1), we only have
to express _(t), _(t) as __(t2) = t2_(t) and __(t2) = _(t) for all positive reals,
see 33.19. Obviously, every such operator is natural and the proposition is
proved. _
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
280 Chapter VII. Further applications
33.23. If we use the invariance with respect to SO(m) in the proof of the above
proposition, we get
Proposition. All _rst order natural F-metrics _ on oriented Riemannian manifolds
of dimensions m form a family parameterized by arbitrary smooth functions
_, _, _, _: (0;1) ! R in the following way. For every Riemannian manifold
(M; g) of dimension m > 3 and tangent vectors u, X, Y 2 TxM
_(M;g)(u)(X; Y ) = _(g(u; u))g(X; Y ) + _(g(u; u))g(u;X)g(u; Y ):
If m = 3 then
_(M;g)(u)(X; Y ) = _(g(u; u))g(X; Y ) + _(g(u; u))g(u;X)g(u; Y )
+ _(g(u; u))g(u;X _ Y )
where _ means the vector product. If m = 2, then
_(M;g)(u)(X; Y ) = _(g(u; u))g(X; Y ) + _(g(u; u))g(u;X)g(u; Y )
+ _(g(u; u))
g(Jg(u);X)g(u; Y ) + g(Jg(u); Y )g(u;X)
_
+ _(g(u; u))
g(Jg(u);X)g(u; Y ) g(Jg(u); Y )g(u;X)
_
where Jg is the canonical almost complex structure on (M; g). In the dimension
m = 1 we get
_(M;g)(u)(X; Y ) = _(g(u; u))g(X; Y ):
33.24. If we combine the results from 33.20{33.23 we deduce that all natural
metrics on tangent bundles of Riemannian manifolds depend on six arbitrary
smooth functions on positive real numbers if m > 1, and on three functions in
dimension one.
The same result remains true for oriented Riemannian manifolds if m > 3
or m = 1, but the metrics depend on 7 real functions if m = 3 and on 10 real
functions in dimension two.
34. Multilinear natural operators
We have already discussed several ways how to _nd natural operators and
all of them involve some results from representation theory. Our general procedures
work without any linearity assumption and we also used them in section
30 devoted to the bilinear operators of the type of Frolicher-Nijenhuis bracket.
However, there are very e_ective methods involving much more linear representation
theory of the jet groups in question which enable us to solve more general
classes of problems concerning linear geometric operations.
In fact, the representation theory of the Lie algebras of the in_nite jet groups,
i.e. the formal vector _elds with vanishing absolute terms, plays an important
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 281
role. Thus, the methods di_er essentially if these Lie algebras have _nite dimension
in the geometric category in question. The best known example beside the
Riemannian manifolds is the category of manifolds with conformal Riemannian
structure.
Although we feel that this theory lies beyond the scope of our book, we would
like to give at least a survey and a sort of interface between the topics and the
terminology of this book and some related results and methods available in the
literature. For a detailed survey on the subject we recommend [Kirillov, 80,
pp. 3{29]. The linear natural operators in the category of conformal pseudo-
Riemannian manifolds are treated in the survey [Baston, Eastwood, 90].
Some basic concepts and results from representation theory were treated in
section 13.
34.1. Recall that every natural vector bundles E1; : : : ;Em;E: Mfn ! FM
of order r correspond to Gr
n-modules V1; : : : ; Vm; V . Further, m-linear natural
operators D: C1(E1 _ _ _ _ _ Em) = C1(E1) _ : : : _ C1(Em) ! C1(E) are
of some _nite order k (depending on D), cf. 19.9, and so they correspond to
m-linear Gk+r
n -equivariant mappings D de_ned on the product of the standard
_bers Tk
nVi of the k-th prolongations JkEi, D: Tk
nV1 _ : : : _ Tk
nVm ! V , see
14.18 or 18.20. Equivalently, we can consider linear Gk+r
n -equivariant maps
D: Tk
nV1 _ _ _ Tk
nVm ! V . We can pose the problem at three levels.
First, we may _x all bundles E1; : : : ;Em;E and ask for all m-linear operators
D: E1 _ _ _ _ _ Em E. This is what we always have done.
Second, we _x only the source E1__ _ __Em, so that we search for all m-linear
geometric operations with the given source. The methods described below are
e_cient especially in this case.
Third, both the source and the target are not prescribed.
We shall _rst proceed in the latter setting, but we derive concrete results only
in the special case of _rst order natural vector bundles and m = 1. Of course, the
results will appear in a somewhat implicit way, since we have to assume that the
bundles in question correspond to irreducible representations of G1
n = GL(n).
We do not lose much generality, for all representations of GL(n) are completely
reducible, except the exceptional indecomposable ones (cf. [Boerner, 67, chapter
V]). But although all tensorial representations are decomposable, it might be a
serious problem to _nd the decompositions explicitly in concrete examples. This
also concerns our later discussion on bilinear operations. In particular, we do
not know how to deduce explicitly (in some short elementary way) the results
from section 30 from the more general results due P. Grozman, see below.
34.2. Given linear representations _, _ of a connected Lie group G on vector
spaces V , W, we know that a linear mapping ': V ! W is a G-module homomorphism
if and only if it is a g-module homomorphism with respect to the
induced representations T_, T_ of the Lie algebra g on V , W, see 5.15. So if
we _nd all gk+r
n -module homomorphisms D: Tk
nV1 _ _ _ Tk
nVm ! V , we describe
all (Gk+r
n )+-equivariant maps and so all operators natural with respect to
orientation preserving di_eomorphisms. Hence we shall be able to analyze the
problem on the Lie algebra level. But we _rst continue with some observations
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
282 Chapter VII. Further applications
concerning the Gr+k
n -modules.
Recall that for every Gr
n-module V with homogeneous degree d (as a G1
n-
module) the induced Gr+k
n -module Tk
nV decomposes as GL(n)-module into the
sum Tk
nV = V0 __ _ __Vk of GL(n)-modules Vi with homogeneous degrees di.
Hence given an irreducible G1
n-module W and a Gk+r
n -module homomorphism
': Tk
nV ! W such that ker' does not include Vk, W must have homogeneous
degree dk and Tk
nV is a decomposable Gr+k
n -module by virtue of 13.14. Hence
in order to _nd all Gk+r
n -module homomorphisms with source Tk
nV we have to
discuss the decomposability of this module. Note Tk
nV is always reducible if
k > 0, cf. 13.14. A corollary in [Terng, 78, p. 812] reads
If V is an irreducible G1
n-module, then Tk
nV is indecomposable except V =
_pRn_, k = 1.
So an explicit decomposition of T1
n(_pRn_) leads to
Theorem. All non zero linear natural operators D: E1 E between two natural
vector bundles corresponding to irreducible G1
n-modules are
(1) E1 = _pT_, E = _p+1T_, D = k:d, where k 2 R, n > p _ 0
(2) E1 = E, D = k:id, k 2 R.
This theorem was originally formulated by J. A. Schouten, partially proved
by [Palais, 59] and proved independently by [Kirillov, 77] and [Terng, 78]. Terng
proved this result by direct (rather technical) considerations and she formulated
the indecomposability mentioned above as a consequence. Her methods are not
suitable for generalizations to m-linear operations or to more general categories
over manifolds.
34.3. If we pass to the Lie algebra level, we can include more information extending
the action of gk+r
n to an action of the whole algebra g = g1 _ g0 _ : : :
of formal vector _elds X =
P1
j_j=0 aj
_x_ @
@xj on Rn. In particular, the action of
the (abelian) subalgebra of constant vector _elds g1 will exclude the general
reducibility of Tk
nV .
Lemma. The induced action of gk+r
n on Tk
nV = (JkE)0Rn is given by the Lie
di_erentiation jr+k
0 X:jk
0 s = jk
0 (LXs) and this formula extends the action to
the Lie algebra g of formal vector _elds. Every gk+r
n -module homomorphism
': Tk
nV ! W is a g-module homomorphism.
Proof. We have
jr+k
0 X:jk
0 s = @
@t
__
0 `expt:jr+k
0 X(jk
0 s) = (by 13.2)
= @
@t
__
0 `jr+k
0 FlXt
(jk
0 s) = (by 14.18)
= @
@t
__
0 jk
0 (E(FlXt
) _ s _ FlX
t) = (by 6.15)
= jk
0
LXs
Each gk+r
n -module homomorphism ': Tk
nV ! W de_nes an operator D natural
with respect to orientation preserving local di_eomorphisms. It follows from 6.15
that every natural linear operator commutes with the Lie di_erentiation (this
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 283
can be seen easily also along the lines of the above computation and we shall
discuss even the converse implication in chapter XI). Hence for all j1
0 X 2 g,
jk
0 s 2 Tk
nV
j1
0 X:'(jk
0 s) = LXDs(0) = D(LXs)(0) = '(jk
0 (LXs)) = '(j1
0 X:jk
0 s)
and so ' is a g-module homomorphism. _
34.4. Consider a Gr
n-module V , a g-module homomorphism ': Tk
nV ! W and
its dual '_ : W_ ! (Tk
nV )_. If W is a Gq
n-module, then the subalgebra bq =
gq _ gq+1 _ : : : in g acts trivially on the image Im'_ _ (Tk
nV )_.
We say that a g-module V is of height p if gq:V = 0 for all q > p and gp:V 6= 0.
De_nition. The vectors v 2 Tk
nV with trivial action of all homogeneous components
of degrees greater then the height of V are called singular vectors.
An analogous de_nition applies to subalgebras a _ g with grading and amodules.
So the linear natural operations between irreducible _rst order natural vector
bundles are described by gk+1
n -submodules of singular vectors in (Tk
nV )_.
Similarly we can treat m-linear operators on replacing (Tk
nV )_ by (Tk
nV1)_
_ _ _ (Tk
nVm)_. Since all modules in question are _nite dimensional, it su_ces
to discuss the highest weight vectors (see 34.8) in these submodules which can
also lead to the possible weights of irreducible modules V . For this purpose, one
can use the methods developed (for another aim) by Rudakov. Remark that the
Kirillov's proof of theorem 34.2 also analyzes the possible weights of the modules
V , but by discussing the possible eigen values of the Laplace-Casimir operator.
First we have to derive some suitable formula for the action of g on (Tk
nV )_.
In what follows, V and W will be G1
n-modules and we shall write @i = @
@xi
2 g1.
34.5. Lemma. (Tk
nV )_ =
Pk
i=0 Si(g1) V _.
Proof. Every multi index _ = i1 : : : ij_j, i1 _ _ _ _ _ ij_j, yields the linear map
`_ : Tk
nV ! V; `_(jk
0 s) = (L@i1
_ : : : _ L@ij_j
s)(0):
Since the elements in g1 commute, we can view the elements in Sj_j(g1) as
linear combinations of maps `_. Now the contraction with V _ yields a linear
map
Pk
i=0 Si(g1) V _ ! (Tk
nV )_. This map is bijective, since (Tk
nV )_ has a
basis induced by the iterated partial derivatives which correspond to the maps
`_. _
This identi_cation is important for our computations. Let us denote `i =
L@i
2 g_
1 = S1(g1), so the elements `_ can be viewed as `_ = `i1
_: : :_`ij_j
2
Sj_j(g1) and we have `_ = 0 if j_j > k. Further, for every ` 2 g we shall denote
ad`_:` = (1)j_j[@i1 ; [: : : [@ij_j ; `] : : : ]].
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
284 Chapter VII. Further applications
34.6. Lemma. The action of ` 2 gq on `_ v 2 Sp V _ is
`:`_ v =
X
_+=_
jj=q
`_ (ad`:`):v +
X
_+=_
jj=q+1
`_ _ (ad`:`)
_
v:
Proof. We compute with ` = jk
0X 2 gq
`:(`_ v)(jk
0 s) = (`_ v)(`:jk
0 s) = (`_ v)(jk
0 (LXs)) = h(`_ _ LXs)(0); vi:
Since `j _ LY = LY _ `j + L
[@j;Y ] for all Y 2 g, 1 _ j _ n, and [@j ; gl] _ gl1,
we get
`:(`_ v)(jk
0 s) = h`i1 : : : `ip1
LX`ips(0); vi + h`i1 : : : `ip1
L
[@ip;X]s(0); vi
and the same procedure can be applied p times in order to get the Lie derivative
terms just at the left hand sides of the corresponding expressions. Each choice
of indices among i1; : : : ; ip determines just one summand of the outcome. Hence
we obtain (the sum is taken also over repeating indices)
`:(`_ v)(jk
0 s) =
X
_+=_
h(ad`:`):`_s(0); vi:
Further ad`:` = 0 whenever jj > q+1 and for all vector _elds Y 2 g0_g1_: : :
we have
h(LY _ `_s)(0); vi = h(`_s)(0);LY vi
so that only the terms with jj = q or jj = q +1 can survive in the sum (notice
Y 2 gp, p _ 1, implies LY v = 0). Since ` = jk
0 Y 2 g0 acts on (the jet of constant
section) v by `:v = LY v(0), we get the result. _
34.7. Example. In order to demonstrate the computations with this formula,
let us now discuss the linear operations in dimension one.
We say that V is a gk
n-module homogeneous in the order if there is k0 such
that gk0 :v = 0 implies v = 0 and gl:v = 0 for all v and l > k0. Each gk
n-module
includes a submodule homogeneous in order. Indeed, the isotropy algebra of
each vector v contains some kernel bl, l _ k, denote lv the minimal one. Let p
be the minimum of these l's. Then the set of vectors with lv = p is a submodule
homogeneous in order. In particular, every irreducible module is homogeneous
in order.
Consider a g11
module V homogeneous in order. For every non zero vector
a = `p
1
v 2 Sp(g1) V _ _ (Tk
1 V )_ and ` 2 g1 we get
`:a = p`p1
1
[@1; `]v +
p
2
_
`p2
1
_ [@1; [@1; `]] v:
Take ` = x2 d
dx so that [@1; `] = 2x d
dx =: 2h and [@1; [@1; `]] = 2@1.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 285
Assume now b1:a = 0. Then
0 = `:a = `p1
1
(2ph:v + p(p 1)v)
so that 2h:v = (p 1)v or p = 0.
Further, set ` = x3 d
dx . We get
0 = `:a =
p
2
_
`p2
1
[@1; [@1; `]]:v
_____p
3
_
`p3
1 :[@1; [@1; [@1; `]]] v
= `p2
1
(3p(p 1)h:v p(p 1)(p 2)v):
Hence either p = 0 or p = 1 or 3h:v = 3
2 (p 1)v = (p 2)v. The latter is not
possible, for it says p = 1. The case p = 0 is not interesting since the action
of b1 on all vectors in V _ = S0(g1)V _ is trivial. But if p = 1 we get h:v = 0
and so the homogeneity in order implies the action of g11
on V is trivial. Hence
V = R if irreducible. Moreover, the submodule generated by a in (T1
1 R)_ is
`1 R with the action h:t`1 = 0 + t`1. Hence if ': T1
1 V ! W is a g-module
homomorphism and if both V and W are irreducible, then either ' factorizes
through ': V ! W which means V = W, ' = k:idV , or V = R, W = R_ with
the minus identical action of g11
. In this way we have proved theorem 34.2 in the
dimension one.
34.8. The situation in higher dimensions is much more di_cult. Let us mention
some concepts and results from representation theory. Our source is [Zhelobenko,
Shtern, 83] and [Naymark, 76].
Consider a Lie algebra g and its representation _ in a vector space V . An
element _ 2 g_ is called a weight if there is a non zero vector v 2 V such that
_(x)v = _(x)v for all x 2 g. Then v is called a weight vector (corresponding
to _). If h _ g is a subalgebra, then the weights of the adjoint representation
of h in g are called roots of the algebra g with respect to h. The corresponding
weight vectors are called the root vectors (with respect to h).
A maximal solvable subalgebra b in a Lie algebra g is called a Borel subalgebra.
A maximal commutative subalgebra h _ g with the property that all operators
adx, x 2 h, are diagonal in g is called a Cartan subalgebra.
In our case g = gl(n), the upper triangular matrices form the Borel subalgebra
b+ while the diagonal matrices form the Cartan subalgebra h. Let us denote
n+ the derived algebra [b+; b+], i.e. the subalgebra of triangular matrices with
zeros on the diagonals. Consider a gl(n)-module V . A vector v 2 V is called
the highest weight vector (with respect to b+) if there is a root _ 2 h_ such that
x:v _(x)v = 0 for all x 2 h and x:v = 0 for all x 2 n+. The root _ is called
the highest weight. In our case we identify h_ with Rn.
The highest weight vectors always exist for complex representations of complex
algebras and are uniquely determined for the irreducible ones. The procedure
of complexi_cation allows to use this for the real case as well. So each
_nite dimensional irreducible representation of gl(n) is determined by a highest
weight (_1; : : : ; _n) 2 C such that all _i _i+1 are non negative integers,
i = 1; : : : ; n 1.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
286 Chapter VII. Further applications
34.9. Examples. Let us start with the weight of the canonical representation
on Rn corresponding to the tangent bundle T. The action of a = (akl
), akl
= _k
j _i
l
for some j < i, (corresponding to the action of X = xi @
@xj given by the negative
of the Lie derivative) on a highest weight vector v must be zero, so that only its
_rst coordinate can be nonzero. Hence the weight is (1; 0; : : : ; 0).
Now we compute the weights of the irreducible modules _pRn_. The action
of X = xi @
@xj on a (constant) form ! is LX!. Since LXdxl = _lj
dxi we
get (cf. 7.6) that if X:! = 0 for all j < i then ! is a constant multiple of
dxnp+1 ^ _ _ _ ^ dxn. Further, the action of Lxi=@xi on dxi1 ^ _ _ _ ^ dxip is
minus identity if i appears among the indices ij and zero if not. Hence the
highest weight is (0; : : : ; 0;1; : : : ;1) with n p zeros. Similarly we compute
the highest weight of the dual _pRn, (1; : : : ; 1; 0; : : : ; 0) with n p zeros.
Analogously one _nds that the highest weight vector of SpRm_ is the symmetric
tensor product of p copies of dxn and the weight is (0; : : : ; 0;p).
34.10. Let us come back to our discussion on singular vectors in (Tk
nV )_ for an
irreducible gl(n)-module V . In our preceding considerations we can take suitable
subalgebras with grading instead of the whole algebra g of formal vector _elds.
It turns out that one can describe in detail the singular vectors in dimension two
and for the subalgebra of divergence free formal vector _elds. We shall denote
this algebra by s(2) and we shall write sr
2 for the Lie algebras of the corresponding
jet groups. We shall not go into details here, they can be found in [Rudakov, 74,
pp. 853{859]. But let us indicate why this description is useful. A subalgebra
a _ g is called a testing subalgebra if there is an isomorphism s(2) ! a _ g
of algebras with gradings and a distinguished subspace w(a) _ g1 such that
g1 = a1 _ w(a), [a;w(a)] = 0.
Lemma. Let V be a g1
n-module, (Tk
nV )_ =
Pk
i=0 Si(g1) V _ and a _ g be a
testing subalgebra. Then _ V =
Pk
i=0 Si(w(a)) V _ _ (Tk
nV )_ is an a0-module
and there is an a-module isomorphism (Tk
nV )_ !
Pk
i=0 Si(a1) _ V onto the
image.
Proof. _ V =
P1
i=0 Si(w(a)) V _ is an a0-module, for [a;w(a)] = 0. We have
P1
i=0 Si(a1) _ V =
P1
i=0 Si(a1)
P1
j=0 Sj(w(a)) V _
=
P1
i=0 Si(a1 _ w(a)) V _: _
34.11. It turns out that there are enough testing subalgebras in the algebra of
formal vector _elds. Using the results on s(2), Rudakov proves that for every g1
n-
module V the homogeneous singular vectors can appear only in V _ _S1(g1)
V _ _ (Tk
nV )_. This is equivalent to the assertion that all linear natural operators
are of order one.
Let us remark that this was also proved by [Terng, 78] in a very interesting
way. She proved that every tensor _eld is locally a sum of _elds with polynomial
coe_cients of degree one in suitable coordinates (di_erent for each summand)
and so the naturality and linearity imply that the orders must be one.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 287
34.12. Now, we know that if there is a homogeneous singular vector x which
does not lie in V _ _ (T1
nV )_ then there must be a highest weight singular vector
x 2 g_
1
V _, for all linear representations in question are _nite dimensional.
Let us write x =
Pk
i=1 li ui, where k _ n and all `i are assumed linearly
independent.
Proposition. Let x =
Pk
i=1 li ui be a singular vector of highest weight with
respect to the Borel algebra b+ _ g1
n. If ui 6= 0, i = 1; : : : ; p, and up+1 = 0,
then ui = 0, i = p + 1; : : : ; k, and up is a highest weight vector with weight
_ = (1; : : : ; 1; 0; : : : ; 0) with n p + 1 zeros. Then the weight of x is _ =
(1; : : : ; 1; 0; : : : ; 0) with n p zeros.
Proof. Since x is singular, we have for all k, j, l (we do not use summation
conventions now)
(1)
0 = xkxl @
@xj :
P
p `p up =
P
p 1 [ @
@xp ; xkxl @
@xj ]:up = xl @
@xj :uk + xk @
@xj :ul:
In particular, for all k, j
xk @
(2) @xj :uk = 0
xj @
@xj :uk = xk @
@xj (3) :uj :
Further, x is a highest weight vector with weight _ = (_1; : : : ; _n) and for all i,
j we have
xi @
@xj :x =
P
p `p xi @
@xj :up +
P
p[ @
@xp ; xi @
@xj (4) ] up
=
P
p `p xi @
@xj :up + `j ui:
If i > j, we have xi @
@xj :x = 0 and so
xi @
@xj (5) :up = 0 p 6= j
xi @
@xj (6) :uj = ui:
Further, xi @
@xi :x = _ix and so (4) implies for all p, i
(7) xi @
@xi :up = (_i _p
i )up:
The latter formula shows that the vectors up are either zero or root vectors
of V _ with respect to the Cartan algebra h with weights _(p) = (_1; : : : ; _n),
_i = _i _p
i . Formula (2) implies that either up = 0 or _p = 1. If up = 0,
then all ul = 0, l _ p, by (6). Assume up 6= 0 and up+1 = 0, i.e. _i = 1, i _ p.
Then (5) and (6) show that up is a highest weight vector. By (3), xj @
@xj :uk =
_(k)juk = xk @
@xj :uj , so that for k = p, j > p, (7) implies _(p)jup = _j:up =
xp @
@xj :uj = 0. Hence _i = 1, i = 1; : : : ; p, and _i = 0, i = p + 1; : : : ; n. _
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
288 Chapter VII. Further applications
34.13. Now it is easy to prove theorem 34.2. If D: E1 E is a linear natural
operator between bundles corresponding to irreducible G1
n-modules V , W, then
either V _ = _pRn, p = 0; : : : ; n 1, and W_ = _p+1Rn, or D is a zero order
operator. The dual of the inclusion W_ ! (T1
nV )_ corresponds to the exterior
di_erential up to a constant multiple.
Let us remark, that the only part of the proof we have not presented in detail
is the estimate of the order, but we mentioned a purely geometric way how to
prove this, cf. 34.11. It might be useful in concrete situations to combine some
general methods with _nal computations in the above style.
34.14. The method of testing subalgebras is heavily used in [Rudakov, 75] dealing
with subalgebras of divergence free formal vector _elds and Hamiltonian vector
_elds. The aim of all the mentioned papers by Rudakov is the study of in_nite
dimensional representations of in_nite dimensional Lie algebras of formal vector
_elds. His considerations are based on the study of the space
P1
i=0 Si(g1)V _
and so the results are relevant for our purposes as well. We should remark that in
the cited papers the action slightly di_ers in notation and the vector _elds xi @
@xj
are identi_ed with the transposed matrix (ai
j) to our (aj
i ) and so the weights correspond
to the Borel subalgebra of lower triangular matrices. Due to Rudakov's
results, a description of all linear operations natural with respect to unimodular
or symplectic di_eomorphisms is also available. In the unimodular case we get
the following result. We write S`n for the category of n-dimensional manifolds
with _xed volume forms and local di_eomorphisms preserving the distinguished
forms.
Theorem. All non zero linear natural operators D: E1 E between two _rst
order natural bundles on category S`n corresponding to irreducible representations
of the _rst order jet group are
(1) E1 = E, D = k:id, k 2 R
(2) E1 = _pT_, E = _p+1T_, D = k:d, k 2 R, n > p _ 0
(3) E1 = _n1T_, E = _1T_, D = k:(d _ i _ d) : _n1T_ ! _nT_ i !
_=
_0T_ !
_1T_, k 2 R.
Let us point out that this theorem describes all linear natural operations not
only up to decompositions into irreducible components but also up to natural
equivalences. For example, to _nd linear natural operations with vector _elds
we have to notice Rn _= _n1Rn_, @
@xp
7! i( @
@xp )(dx1 ^ _ _ _ ^ dxn). Hence the
Lie di_erentiation of the distinguished volume forms corresponds to the exterior
di_erential on (n 1)-forms, the identi_cation of n-forms with functions
yields the divergence of vector _elds and the exterior di_erential of the divergence
represents the `composition' of exterior derivatives from point (3). Beside
the constant multiples of identity, there are no other linear operations (with
irreducible target).
We shall not describe the Hamiltonian case. We remark only that then not
even the di_erential forms correspond to irreducible representations and that
the interesting operations live on irreducible components of them.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 289
34.15. Next we shall shortly comment some results concerning m-linear operations.
We follow mainly [Kirillov, 80]. So __ will denote a representation dual
to a representation _ of G1
n
+ and we write _ for the one-dimensional representation
given by a 7! deta1. Further _(M) is the space of all smooth sections
of the vector bundle E_ over M corresponding to _. In particular _(M) coincides
with nM. To every representation _ we associate the representation
~_ := __ _. The pointwise pairing on _(M) _ __ (M) gives rise to a bilinear
mapping _(M) _ ~_(M) ! n(M), a natural bilinear operation of order zero.
Given two sections s 2 _(M), ~s 2 ~_(M) with compact supports we can integrate
the resulting n-form, let us write hs; ~si for the outcome. We have got a
bilinear functional invariant with respect to the di_eomorphism group Di_M.
For every m-linear natural operator D of type (_1; : : : ; _m; _) we de_ne an
(m + 1)-linear functional
FD(s1; : : : ; sm; sm+1) = hD(s1; : : : ; sm); sm+1i;
de_ned on sections si 2 _i (M), i = 1; : : : ;m, sm+1 2 ~_(M) with compact
supports. The functional FD satis_es
(1) FD is continuous with respect to the C1-topology on _i and ~_
(2) FD is invariant with respect to Di_M
(3) FD = 0 whenever \m+1
i=1 suppsi = ;.
We shall call the functionals with properties (1){(3) the invariant local functionals
of the type (_1; : : : ; _m; ~_).
Theorem. The correspondence D 7! FD is a bijection between the m-linear
natural operators of type (_1; : : : ; _m; _) and local linear functionals of type
(_1; : : : ; _m; ~_).
The proof is sketched in [Kirillov, 80] and consists in showing that each such
functional is given by an integral operator the kernel of which recovers the natural
m-linear operator.
34.16. The above theorem simpli_es essentially the discussion on m-linear natural
operations. Namely, there is the action of the permutation group _m+1
on these operations de_ned by (_FD)(s1; : : : ; sm+1) = FD(s_1; : : : ; s_(m+1)),
_ 2 _m+1. Hence a functional of type (_1; : : : ; _m; _) is transformed into a
functional of type (__1(1); : : : ; __1(m+1)) and so for every operation D of the
type (_1; : : : ; _m; _) there is another operation _D. If _(m + 1) = m + 1, then
this new operation di_ers only by a permutation of the entries but, for example,
if _ transposes only m and m + 1, then _D is of type (_1; : : : ; _m1; ~_; ~_m).
In the simplest case m = 1, the exterior derivative d: pM ! p+1M is
transformed by the only non trivial element in _2 into d: np1M ! npM.
If m = 2, the action of _3 becomes signi_cant. We shall now describe all
operations in this case. Those of order zero are determined by projections of
_1 _2 onto irreducible components.
34.17. First order bilinear natural operators. We shall divide these operations
into _ve classes, each corresponding to some intrinsic construction and
the action of _3.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
290 Chapter VII. Further applications
1. Write _ for the canonical representation of G1
n
+ on Rn, i.e. _ (M) are the
smooth vector _elds on M. For every representation _ we have the Lie derivative
L: _ (M) _ _(M) ! _(M), a natural operation of type (_; _; _). The action
of _3 yields an operation of type (_; ~_; ~_ ) allowing to construct invariantly a
covector density from any two tensor _elds which admit a pointwise pairing into
a volume form. This operation appears often in the lagrangian formalism and
Nijenhuis called it the lagrangian Schouten concomitant.
2. This class contains the operations of the types (_k_ __;_l_ __;_m_
__), where k, l, m are certain integers between zero and n while _, _, _ are
certain complex numbers.
Assume _rst k + l > n + 1. Then an operation exists if m = k + l n 1,
_ = _ + _ 1. Let us choose an auxiliary volume form v 2 _(M) and use the
identi_cation _k_ __ _= _nk_ ___1, i.e. we shall construct an operation of
the type (_k0
_ ___0
;_l0
_ ___0 ;_m0
_ ___0 ) with k0+l0 _ n1, m0 = k0+l0+1
and _0 = _0+_0. Then we can write a _eld of type _k_ __ in the form !:v_1,
! 2 _nkT_M. We de_ne
(1) D(!1:v_1; !2:v_1) = (c1d!1 ^ !2 + c2!1 ^ d!2):v_1
where !1 is a (nk)-form, !2 is a (nl)-form, and c1, c2 are constants. The right
hand side in (1) should not depend on the choice of v. So let us write v = ':~v
where ' is a positive function. Then !1:v_1 = ~!1:~v_1, !2:v_1 = ~!2:~v_1,
with ~!1 = '_1:!1, ~!2 = '_1:!2. After the substitution into (1), there appears
the extra summand
(c1d'_1 ^ !1 ^ '_1!2 + c2'_1!1 ^ d'_1 ^ !2)~vmu1
=
��
(_ 1)c1 + (1)k(_ 1)c2
_
:d(ln') ^ !1 ^ !2:v_1:
Thus (1) is a correct de_nition of an invariant operation if and only if
(2) (_ 1)c1 + (1)k(_ 1)c2 = 0:
Now take k + l _ n + 1. We _nd an operation if and only if m = k + l 1
and _ = _ + _. As before, we _x an auxiliary volume form v and we write
the _elds of type _k_ __ as a:v_ where a is a k-vector _eld. The usual
divergence of vector _elds extends to a linear operation _v on k-vector _elds,
_v(X1 ^_ _ _^Xk) =
Pk
i=1(1)i+1divXi:X1 ^_ _ _ ^i _ _ _^Xk, where ^i means that
the entry is missing. Of course, this divergence depends on the choice of v. We
have
(3)
_'v(X1 ^_ _ _^Xk) = ':_v(X1 ^_ _ _^Xk)+
Xk
i=1
(1)i+1Xi('):X1 ^_ _ _ ^i _ _ _^Xk:
Let us look for a natural operator D of the form
D(a:v_; b:v_) = (c1_v(a) ^ b + c2a ^ _v(b) + c3_v(a ^ b)) :v_:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 291
Formula (3) implies that D is natural if and only if
(4) (_ 1)c1 + (_ + _ 1)c3 = 0; (_ 1)c2 + (_ + _ 1)c3 = 0:
The formulas (2) and (4) de_ne the constants uniquely except the case _ =
_ = 1 when we get two independent operations, see also the _fth class. Let
us point out that the second class involves also the Schouten-Nijenhuis bracket
_pT __qT _p+q1T (the case _ = _ = 0, k+l _ n+1), cf. 30.10, sometimes
also caled the antisymmetric Schouten concomitant, which de_nes the structure
of a graded Lie algebra on the _elds in question. This bracket is given by
[X1 ^ _ _ _ ^ Xk; Y1 ^ _ _ _ ^ Yl]
=
P
i;j(1)i+j [Xi; Yj ] ^ X1 ^ _ _ _ ^i _ _ _ ^ Xk ^ Y1 ^ _ _ _ ^j _ _ _ ^ Yl:
The second class is invariant under the action of _3.
3. The third class is represented by the so called symmetric Schouten concomitant.
This is an operation of type (Sk_; Sl_ ; Sk+l1_ ) with a nice geometric
de_nition. The elements in SkTM can be identi_ed with functions on T_M _berwise
polynomial of degree k. Since there is a canonical symplectic structure on
T_M, there is the Poisson bracket on C1(T_M). The bracket of two _berwise
polynomial functions is also _berwise polynomial and so the bracket gives rise
to our operation.
The action of _3 yields an operation of the type (Sk_; Sl_ __; Slk+1_ __).
If k = 1, this is the Lie derivative and if k = l, we get the lagrangian Schouten
concomitant.
4. This class involves the Frolicher-Nijenhuis bracket, an operation of the type
(_ _k_ _; _ _l_ _; _ _k+l_ _), k+l _ n. The tensor spaces in question are not
irreducible, _ _k_ _ is a sum of _k1_ _ and an irreducible representation _k
of highest weight (1; : : : ; 1; 0; : : : ; 0;1) where 1 appears k-times (the trace-free
vector valued forms). The Frolicher-Nijenhuis bracket is a sum of an operation
of type (_k; _l; _k+l) and several other simpler operations.
If we apply the action of _3 to the Frolicher-Nijenhuis bracket, we get an
operation of the type (_ _m_ _; _ _ _k_ _; _ _ _k+m_ _) which is expressed
through contractions and the exterior derivative.
5. Finally, there are the natural operations which reduce to compositions of
wedge products and exterior di_erentiation. Such operations are always de_ned
if at least one of the representations _1, _2, or one of the irreducible components
of _1_2 coincides with _k_ _. Since]_k_ _ = _nk_ _, this class is also invariant
under the action of _3.
In [Grozman, 80b] we _nd the next theorem. Unfortunately its proof based
on the Rudakov's algebraic methods is not available in the literature. In an
earlier paper, [Grozman, 80a], he classi_ed the bilinear operations in dimension
two, including the unimodular case.
34.18. Theorem. All natural bilinear operators between natural bundles corresponding
to irreducible representations of GL(n) are exhausted by the zero
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
292 Chapter VII. Further applications
order operators, the _ve classes of _rst order operators described in 34.17, the
operators of second and third order obtained by the composition of the _rst and
zero order operators and one exceptional operation in dimension n = 1, see the
example below.
In particular, there are no bilinear natural operations of order greater then
three.
34.19. Example. A tensor density on the real line is determined by one complex
number _, we write f(x)(dx)_ 2 C1(E_R) for the corresponding _elds of
geometric objects. There is a natural bilinear operator D: E2=3
_E2=3 E5=3
D(f(dx)2=3; g(dx)2=3) =
_
2
___
f g
d3f=dx3 d3g=dx3
___
+ 3
___
df=dx dg=dx
d2f=dx2 d2g=dx2
___
_
:(dx)5=3
This is a third order operation which is not a composition of lower order ones.
34.20. The multilinear natural operators are also related to the cohomology
theory of Lie algebras of formal vector _elds. In fact these operators express
zero dimensional cohomologies with coe_cients in tensor products of the spaces
of the _elds in question, see [Fuks, 84]. The situation is much further analyzed
in dimension n = 1 in [Feigin, Fuks, 82]. In particular, they have described all
skew symmetric operations E_ _ _ _ _ _ E_ E_. They have deduced
Theorem. For every _ 2 C, m > 0, k 2 Z, there is at most one skew symmetric
operation D: _mC1E_ ! C1E_ with _ = m_1
2m(m1)k, up to a constant
multiple. A necessary and su_cient condition for its existence is the following:
either k=0, or 0 < k _ m and _ satis_es the quadratic equation
(_ + 1
2 )(k1 + 1) m
_
(_ + 1
2 )(k2 + 1) m
_
= 1
2 (k2 k1)2
with arbitrary positive k1 2 Z, k2 2 Z, k1:k2 = k.
The operator corresponding to the _rst possibility k = 0, D: _mC1(E_R) !
C1(Em_1
2m(m1)R), admits a simple expression
f1(dx)_ ^ _ _ _ ^ fm(dx)_ 7!
______
f1 f0
1 ::: f
(m1)
1
f2 f0
2 ::: f
(m1)
2
: : : : : : : : : : : : :
fm f0
m ::: f(m1)
m
______
(dx)m_+1
2m(m1)
Grozman's operator from 34.19 corresponds to the choice m = 2, k = 2,
_ = 2=3, k1 = 2, k2 = 1. The proof of this theorem is rather involved. It
is based on the structure of projective representations of the algebra of formal
vector _elds on the one-dimensional sphere.
34.21. The problem of _nding all natural m-linear operations has been also formulated
for super manifolds. As far as we know, only the linear operations were
classi_ed, see [Bernstein, Leites, 77], [Leites, 80], [Shmelev, 83], but their results
include also the unimodular, and Hamiltonian cases. Some more information is
also available in [Kirillov, 80].
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
34. Multilinear natural operators 293
34.22. The linear natural operations on conformal manifolds. As we
have seen, the description of the linear natural operators is heavily based on the
structure of the subalgebra in the algebra of formal vector _elds which corresponds
to the jet groups in the category in question. If the category involves
very few morphisms, these algebras become small. In particular, they might
have _nite dimensions like in the case of Riemannian manifolds or conformal
Riemannian manifolds. The former example is not so interesting for the following
reasons: Since all irreducible representations of the orthogonal groups are
O(m;R)-invariant irreducible subspaces in tensor spaces, we can work in the
whole category of manifolds in the way demonstrated in section 33. On the
other hand, if we include the so called spinor representations of the orthogonal
group, we get serious problems with the whole setting. However, the second
example is of highest interest for many reasons coming both from mathematics
and physics and it is treated extensively nowadays. Let us conclude this section
with a very short overview of the known results, for more information see the
survey [Baston, Eastwood, 90] or the papers [Baston, 90], [Branson, 85].
Let us write C for the category of manifolds with a conformal Riemannian
structure, i.e. with a distinguished line bundle in S2+
T_M, and the morphisms
keeping this structure. More explicitely, two metrics g, ^g on M are called conformal
if there is a positive smooth function f onM such that ^g = f2g. A conformal
structure is an equivalence class with respect to this equivalence relation. The
conformal structure on M can also be described as a reduction of the _rst order
frame bundle P1M to the conformal group CO(m;R) = R o O(m;R), and the
conformal morphisms ' are just those local di_eomorphisms which preserve this
reduction under the P1'-action. Thus, each linear representation of CO(m;R)
on a vector space V de_nes a bundle functor on C. The category C is not locally
homogeneous, but it is local.
The main di_erence from the situations typical for this book is that there
are new natural bundles in the category C. In fact, we can take any linear
representation of O(m;R) and a representation of the center R _ GL(m;R)
and combine them together. The representations of the center are of the form
(t:id)(v) = tw:v with an arbitrary real number w, which is called the conformal
weight of the representation or of the corresponding bundle functor. Each
tensorial representation of GL(m;R) induces a representation of CO(m;R) with
the conformal weight equal to the di_erence of the number of covariant and
contravariant indices. In particular, the convention for the weight is chosen in
such a way that the bundle of metrics has conformal weight two. If we restrict
our considerations to the tensorial representations, we exclude nearly all natural
linear operators.
Each isometry of a conformal manifold with respect to an arbitrary metric
from the distinguished class is a conformal morphism. Thus, the Riemannian
natural operators described in section 33 can be taken for candidates in the
classi_cation. But the remaining problems are still so di_cult that a general
solution has not been found yet.
Let us mention at least two possibilities how to treat the problem. The
_rst one is to restrict ourselves to locally conformally at manifolds, i.e. we
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
294 Chapter VII. Further applications
consider only a subcategory in C which is admissible in our sense. Thus, the
classi_cation problem for linear operators reduces to a (di_cult) problem from
the representation theory. But what remains then is to distinguish those natural
operators on the conformally at manifolds which are restrictions of natural
operators on the whole category, and to _nd explicite formulas for them. For
general reasons, there must be a universal formula in the terms of the covariant
derivatives, curvatures and their covariant derivatives. The best known example
is the conformal Laplace operator on functions in dimension 4
D = rara +
1
6R
where rara means the operator of the covariant di_erentiation applied twice
and followed by taking trace, and R is the scalar curvature. The proper conformal
weights ensuring the invariance are 1 on the source and 3 on the target.
The _rst summand D0 = rara of D is an operator which is natural on the
functions with the speci_ed weights on conformally at manifolds and the second
summand is a correction for the general case. In view of this example, the
question is how far we can modify the natural operators (homogeneous in the
order and acting between bundles corresponding to irreducible representations of
CO(m;R)) found on the at manifolds by adding some corrections. The answer
is rather nice: with some few exceptions this is always possible and the order
of the correction term is less by two (or more) than that of D0. Moreover, the
correction involves only the Ricci curvature and its covariant derivatives. This
was deduced in [Eastwood, Rice, 87] in dimension four, and in [Baston, 90] for
dimensions greater than two (the complex representations are treated explicitely
and the authors assert that the real analogy is available with mild changes). In
particular, there are no corrections necessary for the _rst order operators, which
where completely classi_ed by [Fegan, 76]. Nevertheless, the concrete formulas
for the operators (_rst of all for the curvature terms) are rarely available.
Another disadvantage of this approach is that we have no information on the
operators which vanish on the conformally at manifolds, even we do not know
how far the extension of a given operator to the whole category is determined.
The description of all linear natural operators on the conformally at manifolds
is based on the general ideas as presented at the begining of this section.
This means we have to _nd the morphisms of g-modules W_ ! (T1
n V )_, where
g is the algebra of formal vector _elds on Rn with ows consisting of conformal
morphisms. One can show that g = o(n + 1; 1), the pseudo-orthogonal algebra,
with grading g = g1_g0_g1 = Rn_co(n;R)_Rn_. The lemmas 34.5 and 34.6
remain true and we see that (T1
n V )_ is the so called generalized Verma module
corresponding to the representation of CO(n;R) on V . Each homomorphism
W_ ! (T1
n V )_ extends to a homomorphism of the generalized Verma modules
(T1
n W)_ ! (T1
n V )_ and so we have to classify all morphisms of generalized
Verma modules. These were described in [Boe, Collingwood, 85a, 85b]. In particular,
if we start with usual functions (i.e. with conformal weight zero), then
all conformally invariant operators which form a `connected pattern' involving
the functions are drawn in 33.18. (The latter means that there are no more
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Remarks 295
operators having one of the bundles indicated on the diagram as the source or
target.) A very interesting point is a general principal coming from the representation
theory (the so called Jantzen-Zuzkermann functors) which asserts that
once we have got such a `connected pattern' all other ones are obtained by a
general procedure. Unfortunately this `translation procedure' is not of a clear
geometric character and so we cannot get the formulas for the corresponding
operators in this way, cf. [Baston, 90]. The general theory mentioned above
implies that all the operators from the diagram in 33.18 admit the extension to
the whole category of conformal manifolds, except the longest arrow 0 ! m.
By the `translation procedure', the same is ensured for all such patterns, but
the question whether there is an extension for the exceptional `long arrows' is
not solved in general. Some of them do extend, but there are counter examples
of operators which do not admit any extension, see [Branson, 89], [Graham, to
appear].
Another more direct approach is used by [Branson, 85, 89] and others. They
write down a concrete general formula in terms of the Riemannian invariants
and they study the action of the conformal rescaling of the metric. Since it is
su_cient to study the in_nitesimal condition on the invariance with respect to
the rescaling of the metric, they are able to _nd series of conformally invariant
operators. But a classi_cation is available for the _rst and second order operators
only.
Remarks
Proposition 30.4 was proved by [Kol_a_r, Michor, 87]. Proposition 31.1 was
deduced in [Kol_a_r, 87a]. The natural transformations Jr ! Jr were determined
in [Kol_a_r, Vosmansk_a, 89]. The exchange map e_ from 32.4 was introduced by
[Modugno, 89a].
The original proof of the Gilkey theorem on the uniqueness of the Pontryagin
forms, [Gilkey, 73], was much more combinatorial and had not used H. Weyl's
theorem. Our approach is similar to [Atiyah, Bott, Patodi, 73], but we do
not need their polynomiality assumption. The Gilkey theorem was generalized
in several directions. For the case of Hermitian bundles and connections see
[Atiyah, Bott, Patodi, 73], for oriented Riemannian manifolds see [Stredder, 75],
the metrics with a general signature are treated in [Gilkey, 75]. The uniqueness of
the Levi-Civit_a connection among the polynomial conformal natural connections
on Riemannian manifolds was deduced by [Epstein, 75]. The classi_cation of the
_rst order liftings of Riemannian metrics to the tangent bundles covers the results
due to [Kowalski, Sekizawa, 88], who used the so called method of di_erential
equations in their much longer proof. Our methods originate in [Slov_ak, 89] and
an unpublished paper by W. M. Mikulski.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
296
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