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CHAPTER XII. GAUGE NATURAL BUNDLES AND OPERATORS
In chapters IV and V we have explained that the natural bundles coincide
with the associated _ber bundles to higher order frame bundles on manifolds.
However, in both di_erential geometry and mathematical physics one can meet
_ber bundles associated to an `abstract' principal bundle with an arbitrary structure
group G. If we modify the idea of bundle functor to such a situation, we
obtain the concept of gauge natural bundle. This is a functor on principal _ber
bundles with structure group G and their local isomorphisms with values in _ber
bundles, but with _bration over the original base manifold. The most important
examples of gauge natural bundles and of natural operators between them are
related with principal connections. In this chapter we _rst develop a description
of all gauge natural bundles analogous to that in chapter V. In particular, we
prove that the regularity condition is a consequence of functoriality and locality
and that any gauge natural bundle is of _nite order. We also present sharp
estimates of the order depending on the dimensions of the standard _bers. So
the r-th order gauge natural bundles coincide with the _ber bundles associated
to r-th principal prolongations of principal G-bundles (see 15.3), which are in
bijection with the actions of the group Wrm
G on manifolds.
Then we discuss a few concrete problems on _nding gauge natural operators.
The geometrical results of section 52 are based on a generalization of the
Utiyama theorem on gauge natural Lagrangians. First we determine all gauge
natural operators of the curvature type. In contradistinction to the essential
uniqueness of the curvature operator on general connections, this result depends
on the structure group in a simple way. Then we study the di_erential forms
of Chern-Weil type with values in an arbitrary associated vector bundle. We
_nd it interesting that the full list of all gauge natural operators leads to a new
geometric result in this case. Next we determine all _rst order gauge natural
operators transforming principal connections to the tangent bundle. In the last
section we _nd all gauge natural operators transforming a linear connection on
a vector bundle and a classical linear connection on the base manifold into a
classical linear connection on the total space.
51. Gauge natural bundles
We are going to generalize the description of all natural bundles F : Mfm !
FMderived in sections 14 and 22 to the gauge natural case. Since the concepts
and considerations are very similar to some previous ones, we shall proceed in a
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
51. Gauge natural bundles 395
rather brief style.
51.1. Let B: FM ! Mf be the base functor. Fix a Lie group G and recall
the category PBm(G), whose objects are principal G-bundles over m-manifolds
and whose morphisms are the morphisms of principal G-bundles f : P ! _ P with
the base map Bf : BP ! B _ P lying in Mfm.
De_nition. A gauge natural bundle over m-dimensional manifolds is a functor
F : PBm(G) ! FM such that
(a) every PBm(G)-object _ : P ! BP is transformed into a _bered manifold
qP : FP ! BP over BP,
(b) every PBm(G)-morphism f : P ! _ P is transformed into a _bered morphism
Ff : FP ! F _ P over Bf,
(c) for every open subset U _ BP, the inclusion i : _1(U) ! P is transformed
into the inclusion Fi : q1
P (U) ! FP.
If we intend to point out the structure group G, we say that F is a G-natural
bundle.
51.2. If two PBm(G)-morphisms f, g : P ! _ P satisfy jr
yf = jr
yg at a point
y 2 Px of the _ber of P over x 2 BP, then the fact that the right translations
of principal bundles are di_eomorphisms implies jr
zf = jr
zg for every z 2 Px. In
this case we write jr
xf = jr
xg.
De_nition. A gauge natural bundle F is said to be of order r, if jr
xf = jr
xg
implies FfjFxP = FgjFxP.
51.3. De_nition. A G-natural bundle F is said to be regular if every smoothly
parameterized family of PBm(G)-morphisms is transformed into a smoothly parameterized
family of _bered maps.
51.4. Remark. By de_nition, a G-natural bundle F : PBm(G) ! FMsatis_es
B _ F = B and the projections qP : FP ! BP form a natural transformation
q : F ! B.
In general, we can consider a category C over _bered manifolds, i.e. C is
endowed with a faithful functor m: C ! FM. If C admits localization of objects
and morphisms with respect to the preimages of open subsets on the bases with
analogous properties to 18.2, we can de_ne the gauge natural bundles on C as
functors F : C ! FMsatisfying B_F = B_m and the locality condition 51.1.(c).
Let us mention the categories of vector bundles as examples. The di_erent way
of localization is the source of a crucial di_erence between the bundle functors
on categories over manifolds and the (general) gauge natural bundles. For any
two _bered maps f, g : Y ! _ Y we write jr
xf = jr
xg, x 2 BY , if jr
yf = jr
yg for
all y 2 Yx. Then we say that f and g have the same _ber r-jet at x. The space
of _ber r-jets between C-objects Y and _ Y is denoted by Jr(Y; _ Y ). For a general
category C over _bered manifolds the _niteness of the order of gauge natural
bundles is expressed with the help of the _ber jets. The description of _nite
order bundle functors as explained in section 18 could be generalized now, but
there appear di_culties connected with the (generally) in_nite dimension of the
corresponding jet groups. Since we will need only the gauge natural bundles
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
396 Chapter XII. Gauge natural bundles and operators
on PBm(G) in the sequel, we will restrict ourselves to this category. Then the
description will be quite analogous to that of classical natural bundles. Some
basic steps towards the description in the general case were done in [Slov_ak, 86]
where the in_nite dimensional constructions are performed with the help of the
smooth spaces in the sense of [Frolicher, 81].
51.5. Examples.
(1) The choice G = feg reproduces the natural bundles on Mfm
(2) The functors Qr : PBm(G) ! FM of r-th order principal connections
mentioned in 17.4 are examples of r-th order regular gauge natural bundles.
(3) The gauge natural bundles Wr : PBm(G) ! PBm(Wrm
G) of r-th principal
prolongation de_ned in 15.3 play the same role as the frame bundles Pr : Mfm !
FM did in the description of natural bundles.
(4) For every manifold S with a smooth left action ` of Wrm
G, the construction
of associated bundles to the principal bundlesWrP yields a regular gauge natural
bundle L: PBm(G) ! FM. We shall see that all gauge natural bundles are of
this type.
51.6. Proposition. Every r-th order regular gauge natural bundle is a _ber
bundle associated to Wr.
Proof. Analogously to the case of natural bundles, an r-th order regular gauge
natural bundle F is determined by the system of smooth associated maps
FP; _ P : Jr(P; _ P) _BP FP ! F _ P
and the restriction of FRm_G;Rm_G to the _ber jets at 0 2 Rm yields an action
of Wrm
G = Jr
0(Rm _ G;Rm _ G)0 on the _ber S = F0(Rm _ G). The same
considerations as in 14.6 complete now the proof. _
51.7. Theorem. Let F : PBm(G) ! Mf be a functor endowed with a natural
transformation q : F ! B such that the locality condition 51.1.(c) holds.
Then S := (qRm_G)1(0) is a manifold of dimension s _ 0 and for every
P 2 ObPBm(G), the mapping qP : FP ! BP is a locally trivial _ber bundle
with standard _ber S, i.e. F : PBm(G) ! FM. The functor F is a regular
gauge natural bundle of a _nite order r _ 2s + 1. If moreover m > 1, then
(1) r _ maxf
s
m 1;
s
m
+ 1g:
All these estimates are sharp.
Briey, every gauge natural bundle on PBm(G) with s-dimensional _bers is
one of the functors de_ned in example 51.5.(4) with r bounded by the estimates
from the theorem depending on m and s but not on G. The proof is based on
the considerations from chapter V and it will require several steps.
51.8. Let us point out that the restriction of any gauge natural bundle F to
trivial principal bundles M _G and to morphisms of the form f _id : M _G !
N _ G can be viewed as a natural bundle Mfm ! FM. Hence the action _ of
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
51. Gauge natural bundles 397
the abelian group of _ber translations tx : Rm_G ! Rm_G, (y; a) 7! (x+y; a),
i.e. _x = Ftx, is a smooth action by 20.3. This implies immediately the assertion
on _ber bundle structure in 51.7, cf. 20.3. Further, analogously to 20.5.(1) we
_nd that the regularity of F follows if we verify the smoothness of the induced
action of the morphisms keeping the _ber over 0 2 Rm on the standard _ber
S = F0(Rm _ G).
51.9. Lemma. Let U _ S be a relatively compact open set and write
QU =
[
'
F'(U) _ S
where the union goes through all ' 2 PBm(G)(Rm _ G;Rm _ G) with '0(0) =
(0). Then there is r 2 N such that for all z 2 QU and all PBm(G)-morphisms
', : Rm _ G ! Rm _ G, '0(0) = 0(0) = 0, the condition jr
0' = jr
0 implies
F'(z) = F (z).
Proof. Every morphism ': Rm_G ! Rm_G is identi_ed with the couple '0 2
C1(Rm;Rm), _' 2 C1(Rm;G). So F induces an operator ~ F : C1(Rm;Rm _
G) ! C1(F(Rm _ G); F(Rm _ G)) which is qRm_G-local and the map qRm_G
is locally non-constant. Consider the constant map ^e: Rm ! G, x 7! e, and the
map idRm _^e: Rm ! Rm_G corresponding to idRm_G. By corollary 19.8, there
is r 2 N such that jr
0f = jr
0(idRm _ ^e) implies ~ Ff(z) = z for all z 2 U. Hence if
jr
0' = jr
0idRm_G, then F'(z) = z for all z 2 U and the easy rest of the proof is
quite analogous to 20.4. _
51.10. Proposition. Every gauge natural bundle is regular.
Proof. The whole proof of 20.5 goes through for gauge natural bundles if we
choose local coordinates near to the unit in G and replace the elements j1
0 fn 2
G1
m by the couples (j1
0 fn; j1
0 _'n) 2 G1
m o T1
m G and idRm by idRm _ ^e. Let us
remark that also _x gets the new meaning of F(tx). _
51.11. Since every natural bundle F : Mf ! FM can be viewed as the gauge
natural bundle _ F = F _ B: PBm(G) ! FM, the estimates from theorem 51.7
must be sharp if they are correct, see 22.1. Further, the considerations from 22.1
applied to our situation show that we complete the proof of 51.7 if we deduce
that every smooth action of Wrm
G on a smooth manifold S factorizes to an action
of Wkm
G, k _ r, with k satisfying the estimates from 51.7.
So let us consider a continuous action _: Wrm
G ! Di_(S) and write H for its
kernel. Hence H is a closed normal Lie subgroup and the kernel H0 _ Gr
m of
the restriction _0 = _jGr
m always contains the normal Lie subgroup Brk
_ Gr
m
with k = 2dimS + 1 if m = 1 and k = maxfdimS
m1 ; dimS
m + 1g if m > 1. Let us
denote Kr
k the kernel of the jet projection Wrm
G ! Wkm
G.
Lemma. For every Lie group G and all r, k 2 N, r > k _ 1, the normal closed
Lie subgroup in Wrm
G generated by Brk
o feg equals to Kr
k.
Proof. The Lie group Wrm
G can be viewed as the space of _ber jets Jr
0(Rm _
G;Rm _G)0 and so its Lie algebra wr
mg coincides with the space of _ber jets at
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
398 Chapter XII. Gauge natural bundles and operators
0 2 Rm of (projectable) right invariant vector _elds with projections vanishing
at the origin. If we repeat the consideration from the proof of 13.2 with jets
replaced by _ber jets, we get the formula for Lie bracket in wr
mg, [jr
0X; jr
0Y ] =
jr
0[X; Y ]. Since every polynomial vector _eld in wr
mg decomposes into a sum
of X1 2 gr
m and a vertical vector _eld X2 from the Lie algebra Trm
g of Trm
G, we
get immediately the action of gr
m on Trm
g, [jr
0X1 + 0; 0 + jr
0X2] = jr
0
LX1X2.
Now let us _x a base ei of g and elements Yi 2 Trm
g, Yi = jr
0x1ei. Taking any
functions fi on Rm with jk
0 fi = 0, the r-jets of the _elds Xi = fi@=@x1 lie in the
kernel br
k
_ gr
m and we get
X
i
[jr
0Xi; jr
0Yi] = jr
0fiei 2 Trm
g:
Hence [br
k; Trm
g] contains the whole Lie algebra of the kernel Kr
k and so the latter
algebra must coincide with the ideal in wr
mg generated by br
k o f0g. Since the
kernel Kr
1 is connected this completes the proof. _
51.12. Corollary. Let G be a Lie group and S be a manifold with a continuous
left action of Wrm
G, dimS = s _ 0. Then the action factorizes to an action of
Wkm
G with k _ 2s + 1. If m > 1, then k _ maxf s
m1 ; s
m + 1g. These estimates
are sharp.
The corollary concludes the proof of theorem 51.7.
51.13. Given two G-natural bundles F, E: PBm(G) ! FM, every natural
transformation T : F ! E is formed be a system of base preserving FMmorphisms,
cf. 14.11 and 51.8. In the same way as in 14.12 one deduces
Proposition. Natural transformations F ! E between two r-th order Gnatural
bundles over m-dimensional manifolds are in a canonical bijection with
the Wrm
G-equivariant maps F0 ! E0 between the standard _bers F0 = F0(Rm_
G), E0 = E0(Rm _ G).
51.14. De_nition. Let F and E be two G-natural bundles over m-dimensional
manifolds. A gauge natural operator D: F E is a system of regular operators
DP : C1FP ! C1EP for all PBm(G)-objects _ : P ! BP such that
(a) D_ P (Ff _ s _ Bf1) = Ff _ DP s _ Bf1 for every s 2 C1FP and every
PBm(G)-isomorphism f : P ! _ P,
(b) D_1(U)(sjU) = (DP s)jU for every s 2 C1FP and every open subset
U _ BP.
51.15. For every k 2 N and every gauge natural bundle F of order r its composition
Jk _ F with the k-th jet prolongation de_nes a gauge natural bundle
functor of order k + r, cf. 14.16. In the same way as in 14.17 one deduces
Proposition. The k-th order gauge natural operators F E are in a canonical
bijection with the natural transformations JkF ! E.
In particular, this proposition implies that the k-th order G-natural operators
F E are in a canonical bijection with the Wsm
G-equivariant maps Jk
0 F ! E0,
where s is the maximum of the orders of JkF and E and Jk
0 F = Jk
0 F(Rm _G).
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
52. The Utiyama theorem 399
51.16. Consider the G-natural connection bundle Q and an arbitrary G-natural
bundle E.
Proposition. Every gauge natural operator A: Q E has _nite order.
Proof. By 51.8, every G-natural bundle F determines a classical natural bundle
NF by NF(M) = F(M_G), NF(f) = F(f _idG). Given a G-natural operator
D: F E, we denote by ND its restriction to NF, i.e. NDM = DM_G. Clearly,
ND is a classical natural operator NF ! NE.
Since our operator A is determined locally, we may restrict ourselves to the
product bundle M _ G. Then we have a classical natural operator NA. In this
situation the standard _ber g Rm_ of Q coincides with the direct product of
dimG copies of Rm_. Hence we can apply proposition 23.5. _
52. The Utiyama theorem
52.1. The connection bundle. First we write the equations of a connection
on Rm _ G in a suitable form. Let ep be a basis of g and let !p be the
corresponding (left) Maurer-Cartan forms given by
P
p !p(Xg)ep = T(_g1 )(X).
Let
(1) (!p)e = p
i (x)dxi
be the equations of (x; e), x 2 Rm, e = the unit of G. Since is right-invariant,
its equations on the whole space Rm _ G are
(2) !p = p
i (x)dxi:
The connection bundle QP = J1P=G is a _rst order gauge natural bundle
with standard _ber gRm_. Having a PBm(G)-isomorphism _ of Rm _G into
itself
(3) _x = f(x); _y = '(x) _ y; f(0) = 0
with ': Rm ! G, its 1-jet j10
_ 2 W1m
G is characterized by
(4) a = '(0) 2 G; (ap
i ) = j1
0 (a1 _ '(x)) 2 g Rm_; (ai
j) = j1
0f 2 G1
m:
Let Apq
(a) be the coordinate expression of the adjoint representation of G. In
15.6 we deduced the following equations of the action of W1m
G on g Rm_
(5) _
p
i = Apq
(a)(q
j + aq
j )~aj
i :
The _rst jet prolongation J1QP of the connection bundle is a second order
gauge natural bundle, so that its standard _ber S1 = J1
0Q(Rm _ G), with the
coordinates p
i , p
ij = @p
i =@xj , is a W2m
G-space. The second order partial
derivatives ap
ij of the map a1 _ '(x) together with ai
jk = @2
jkfi(0) are the additional
coordinates on W2m
G. Using 15.5, we deduce from (5) that the action of
W2m
G on S1 has the form (5) and
_
p
ij = Apq
(a)q
kl~aki
~al
j + Apq
(a)aq
kl~aki
~al
(6) j+
+ Dp
qr(a)q
karl
~aki
~al
j + Ep
qr(a)aq
karl
~aki
~al
j + Apq
(a)(q
k + aq
k)~ak
ij
where the D's and E's are some functions on G, which we shall not need.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
400 Chapter XII. Gauge natural bundles and operators
52.2. The curvature. To deduce the coordinate expression of the curvature
tensor, we shall use the structure equations of . By 52.1.(1), the components
'p of the connection form of are
(1) 'p = !p p
i (x)dxi:
The structure equations of reads
(2) d'p = cpq
r'q ^ 'r + Rp
ijdxi ^ dxj
where cpq
r are the structure constants of G and Rp
ij is the curvature tensor. Since
!p are the Maurer-Cartan forms of G, we have d!p = cpq
r!q ^ !r. Hence the
exterior di_erentiation of (1) yields
(3) d'p = cpq
r('q + q
i dxi) ^ ('r + rj
dxj) + p
ij(x)dxi ^ dxj :
Comparing (2) with (3), we obtain
(4) Rp
ij = p
[ij] + cpq
rq
i rj
:
52.3. Generalization of the Utiyama theorem. The curvature of a connection
on P can be considered as a section CP : BP ! LP _2T_BP, where
LP = P[g; Ad] is the so-called adjoint bundle of P, see 17.6. Using the language
of the theory of gauge natural bundles, D. J. Eck reformulated a classical result
by Utiyama in the following form: All _rst order gauge natural Lagrangians on
the connection bundle are of the form A _ C, where A is a zero order gauge
natural Lagrangian on the curvature bundle and C is the curvature operator,
[Eck, 81]. By 49.1, a _rst order Lagrangian on a connection bundle QP is a
morphism J1QP ! _mT_BP, so that the Utiyama theorem deals with _rst
order gauge natural operators Q _mT_B. We are going to generalize this
result. Since the proof will be based on the orbit reduction, we shall directly
discuss the standard _bers in question.
Denote by : S1 ! g _2Rm_ the formal curvature map 52.2.(4). One
sees easily that is a surjective submersion. The semi-direct decomposition
W2m
G = G2
m o T2m
G together with the target jet projection T2m
G ! G de_nes a
group homomorphism p: W2m
G ! G2
m
_G. Let Z be a G2
m
_G-space, which can
be considered as a W2m
G-space by means of p. The standard _ber g_2Rm_ of
the curvature bundle is a G1
m
_ G-space, which can be interpreted as G2
m
_ Gspace
by means of the jet homomorphism _2
1 : G2
m
! G1
m.
Proposition. For every W2m
G-map f : S1 ! Z there exists a unique G2
m
_ Gmap
g : g _2Rm_ ! Z satisfying f = g _ .
Proof. On the kernel K of p: W2m
G ! G2
m
_ G we have the coordinates ap
i ,
ap
ij = ap
ji introduced in 52.1. Let us replace the coordinates p
ij on S1 by
(1) Rp
ij = p
[ij] + cpq
rq
i rj
; Sp
ij = p
(ij);
while p
i remain unchanged. Hence the coordinate form of is (p
i ;Rp
ij ; Sp
ij) 7!
(Rp
ij ). From 52.1.(5) and 52.1.(6) we can evaluate ap
i and ap
ij in such a way that
_
p
i = 0 and _ Sp
ij = 0. This implies that each _ber of is a K-orbit. Then we
apply 28.1. _
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
52. The Utiyama theorem 401
52.4. To interpret the proposition 52.3 in terms of operators, it is useful to
introduce a more subtle notion of principal prolongation Ws;rP of order (s; r),
s _ r, of a principal _ber bundle P(M;G). Formally we can construct the _ber
product over M
(1) Ws;rP = PsM _M JrP
and the semi-direct product of Lie groups
(2) Ws;r
m G = Gs
m o Trm
G
with respect to the right action (A;B) 7! B _ _s
r(A) of Gs
m on Trm
G. The right
action of Ws;r
m G on Ws;rP is given by a formula analogous to 15.4
(u; v)(A;B) = (u _ A; v:(B _ _s
r(A1 _ u1)));
u 2 PsM, v 2 JrP, A 2 Gs
m, B 2 Trm
G. In the case r = 0 we have a
direct product of Lie groups Ws;0
m G = Gs
m
_ G and the usual _bered product
Ws;0P = PsM _M P of principal _ber bundles.
To clarify the geometric substance of the previous construction, we have to use
the concept of (r; s; q)-jet of a _bered manifold morphism introduced in 12.19.
Then Ws;rP can be de_ned as the space of all (r; r; s)-jets at (0; e) of the local
principal bundle isomorphisms Rm _ G ! P and the group Ws;r
m G is the _ber
of Ws;r(Rm _ G) over 0 2 Rm endowed with the jet composition. The proof is
left to the reader as an easy exercise. Furthermore, in the same way as in 51.2
we deduce that if two PBm(G)-morphisms f; g : P ! _ P satisfy jr;r;s
y f = jr;r;s
y g
at a point y 2 Px, x 2 BP, then this equality holds at every point of the _ber
Px. In this case we write jr;r;s
x f = jr;r;s
x g.
Now we can say that natural bundle F is of order (s; r), s _ r, if jr;r;s
x f =
jr;r;s
x g implies FfjFxP = FgjFxP. Using the proposition 51.10 we deduce quite
similarly to 51.6 that every gauge natural bundle of order (s,r) is a _ber bundle
associated to Ws;r.
Then the proposition 52.3 is equivalent to the following assertion.
General Utiyama theorem. Let F be a gauge natural bundle of order (2; 0).
Then for every _rst order gauge natural operator A: Q F there exists a
unique natural transformation A_: L_2T_B ! F satisfying A = A_ _ C, where
C : Q L _2T_B is the curvature operator.
In all concrete problems in this chapter the result will be applied to gauge
natural bundles of order (1,0). By de_nition, every such a bundle has the order
(2,0) as well.
52.5. Curvature-like operators. The curvature operator C : Q L_2T_B
is a gauge natural operator because of the geometric de_nition of the curvature.
We are going to determine all gauge natural operators Q L 2T_B.
(We shall see that the values of all of them lie in L _2T_B. But this is an
interesting geometric result that the antisymmetry of such operators is a consequence
of their gauge naturality.) Let Z _ L(g; g) be the subspace of all
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
402 Chapter XII. Gauge natural bundles and operators
linear maps commuting with the adjoint action of G. Since every z 2 Z is an
equivariant linear map between the standard _bers, it induces a vector bundle
morphism _zP : LP ! LP. Hence we can construct a modi_ed curvature operator
C(z)P : (_zP _2T_idBP ) _ CP .
Proposition. All gauge natural operators Q L 2T_B are the modi_ed
curvature operators C(z) for all z 2 Z.
Proof. By 51.16, every gauge natural operator A on the connection bundle has
_nite order. The r-th order gauge natural operators correspond to the Wr+1
m Gequivariant
maps Jr
0Q ! g 2Rm_. Let p
i_ be the induced coordinates on
Jr
0Q, where _ is a multi index of range m with j_j _ r. On g 2Rm_ we have
the canonical coordinates Rp
ij and the action
(1) _Rp
ij = Apq
(a)Rq
kl~aki
~al
j :
Hence the coordinate components of the map associated to A are some functions
fp
ij(q
k_). If we consider the canonical inclusion of G1
m into Wr+1
m G, then
analogously to 14.20 the transformation laws of all quantities p
i_ are tensorial.
The equivariance with respect to the homotheties in G1
m gives a homogeneity
condition
(2) c2fp
ij(q
k_) = fp
ij(c1+j_jq
k_) 0 6= c 2 R:
By the homogeneous function theorem, fp
ij is independent of p
i_ with j_j _ 2.
Hence A is a _rst order operator and we can apply the general Utiyama theorem.
The associated map
(3) g : g _2Rm_ ! g 2Rm_
of the induced natural transformation L _2T_B ! L 2T_B is of the form
gp
ij(Rq
kl). Using the homotheties in G1
m we _nd that g is linear. If we _x one
coordinate in g on the right-hand side of the arrow (3), we obtain a linear G1
m-
map _n_2Rm_ ! 2Rm_. By 24.8.(5), this map is a linear combination of the
individual inclusions _2Rm_ ,! 2Rm_, i.e.
(4) gp
ij = zp
qRq
ij :
Using the equivariance with respect to the canonical inclusion of G into W2m
G,
we _nd that the linear map (zp
q ) : g ! g commutes with the adjoint action. _
52.6. Remark. In the case that the structure group is the general linear group
GL(n) of an arbitrary dimension n, the invariant tensor theorem implies directly
that the Ad-invariant linear maps gl(n) ! gl(n) are generated by the identity
and the map X 7! (traceX)id. Then the proposition 52.5 gives a two-parameter
family of all GL(n)-natural operators Q L 2T_B, which the _rst author
deduced by direct evaluation in [Kol_a_r, 87b]. In general it is remarkable that
the study of the case of the special structure group GL(n), to which we can
apply the generalized invariant tensor theorem, plays a useful heuristic role in
the theory of gauge natural operators.
Further we remark that all gauge natural operators Q _ Q L 2T_B
transforming pairs of connections on an arbitrary principal _ber bundle P into
sections of LP 2T_BP are determined in [Kurek, to appear a].
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
52. The Utiyama theorem 403
52.7. Generalized Chern-Weil forms. We recall that for every vector bundle
E ! M, a section of E _rT_M is called an E-valued r-form, see 7.11.
For E = M _ R we obtain the usual exterior forms on M. Consider a linear
action _ of a Lie group G on a vector space V and denote by ~ V the G-natural
bundle over m-manifolds determined by this action of G = W0m
G. We are going
to construct some gauge natural operators transforming every connection on a
principal bundle P(M;G) into a ~ V (P)-valued exterior form. In the special case
of the identity action of G on R, i.e. _(g) = idR for all g 2 G, we obtain the
classical Chern-Weil forms of , [Kobayashi,Nomizu, 69].
Let h: Srg ! V be a linear G-map. We have Sr(g _2Rm_) = Srg
Sr_2Rm_, so that we can de_ne _h : g _2Rm_ ! V _2rRm_ by
(1) _h(A) = (h Alt)(A _ _ _ A); A 2 g _2Rm_;
where Alt: Sr_2Rm_ ! _2rRm_ is the tensor alternation. Since g _2Rm_
or V _2rRm_ is the standard _ber of the curvature bundle or of ~ V (P)
_2rT_M, respectively, _h induces a bundle morphism _hP : L(P) _2T_M !
~ V (P) _2rT_M. For every connection : M ! QP, we _rst construct its
curvature CP and then a ~ V (P)-valued 2r-form
(2) ~hP () =_hP (CP ):
Such forms will be called generalized Chern-Weil forms.
Let I(g; V ) denote the space of all polynomial G-maps of g into V . Every
H 2 I(g; V ) is determined by a _nite sequence of linear G-maps hri : Srig ! V ,
i = 1; : : : ; n. Then
~H
P () =~hr1
P () + _ _ _ +~hrn
P ()
is a section of ~ V (P) _T_M for every connection on P. By de_nition, ~H is
a gauge natural operator Q ~ V _T_B.
52.8. Theorem. All G-natural operators Q ~ V _T_B are of the form ~H
for all H 2 I(g; V ).
Proof. Consider some linear coordinates yp on g and za on V and the induced
coordinates yp
ij on g _2Rm_ and za
i1:::is on V _sRm_.
By 51.16 every G-natural operator A: Q ~ V _sT_B has a _nite order k.
Hence its associated map f : Jk
0Q ! V _sRm_ is of the form
za
i1:::is = fa
i1:::is(p
i_); 0 _ j_j _ k:
The homotheties in G1
m give a homogeneity condition
ksfa
i1:::is(p
i_) = fa
i1:::is (k1+j_jp
i_):
This implies that f is a polynomial map in p
i_. Fix a, p1, j_1j, : : : , pr, j_rj with
j_1j _ 2 and consider the subpolynomial of the a-th component of f which is
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
404 Chapter XII. Gauge natural bundles and operators
formed by the linear combinations of p1
i1_1 : : : pr
ir_r
. It represents a GL(m)-map
Rm_ Sj_1jRm_ _ : : : _ Rm_ Sj_rjRm_ ! _pRm_. Analogously to 24.8 we
deduce that this is the zero map because of the symmetric component Sj_1jRm_.
Hence A is a _rst order operator.
Applying the general Utiyama theorem, we obtain f = g _ , where g is a
G1
m
_ G-map g _2Rm_ ! V _sRm_. The coordinate form of g is
za
i1:::is = ga
i1:::is (yp
ij):
Using the homotheties in G1
m we _nd that s = 2r and g is a polynomial of degree
r in yp
ij . Its total polarization is a linear map Sr(g _2Rm_) ! V _2rRm_.
If we _x one coordinate in V and any r-tuple of coordinates in g, we obtain
an underlying problem of _nding all linear G1
m-maps r_2Rm_ ! _2rRm_. By
24.8.(5) each this map is a constant multiple of yp1
[i1i2
: : : ypr
i2r1i2r]. Hence g is of
the form
cap
1:::pryp1
[i1i2
: : : ypr
i2r1i2r]:
The equivariance with respect to the canonical inclusion of G into W2m
G implies
that (cap
1:::pr ) : Srg ! V is a G-map. _
52.9. Consider the special case of the identity action of G on R. Then every
linear G-map Srg ! R is identi_ed with a G-invariant element of Srg_ and
the (M _ R)-valued forms are the classical di_erential forms on M. Hence
52.7.(2) gives the classical Chern-Weil forms of a connection. In this case the
theorem 52.8 reads that all gauge natural di_erential forms on connections are
the classical Chern-Weil forms. All of them are of even degree. The exterior
di_erential of a Chern-Weil form is a gauge natural form of odd degree. By the
theorem 52.8 it must be a zero form. This gives an interesting application of
gauge naturality for proving the following classical result.
Corollary. All classical Chern-Weil forms are closed.
52.10. In general, if one has a vector bundle E ! M, an E-valued r-form
!: _rTM ! E and a linear connection _ on E, one introduces the covariant
exterior derivative d_!: _r+1TM ! E, see 11.14. Consider the situation from
52.7. For every H 2 I(g; V ) and every connection on P we have constructed
a ~ V (P)-valued form ~HP (), which is of even degree. According to 11.11,
induces a linear connection V on ~ V (P). Then dV
~H
P () is a gauge natural
~ V (P)-valued form of odd degree. By the theorem 52.8 it is a zero form. Thus,
we have proved the following interesting geometric result.
Proposition. For every H 2 I(g; V ) and every connection on P, it holds
dV
~H
P () = 0.
52.11. Remark. We remark that another generalization of Chern-Weil forms
is studied in [Lecomte, 85].
52.12. Gauge natural approach to the Bianchi identity. It is remarkable
that the Bianchi identity for a principal connection : BP ! QP can be deduced
in a similar way. Using the notation from 52.5, we _rst prove an auxiliary result.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
53. Base-extending gauge natural operators 405
Lemma. The only gauge natural operator Q L_3T_B is the zero operator.
Proof. By 51.16, every such operator A has _nite order. Let
fp
ijk(q
l_); 0 _ j_j _ r
be its associated map. The homotheties in G1
m yield a homogeneity condition
(1) c3fp
ijk(q
l_) = fp
ijk(c1+j_jq
l_); c 2 R n f0g:
Hence f is polynomial in p
i , p
ij and p
ijk of degrees d0, d1 and d2 satisfying
3 = d0 + 2d1 + 3d2:
This implies f is linear in p
ijk. But p
ijk represent a linear GL(m)-map Rm_
S2Rm_ ! _3Rm_ for each p = 1; : : : ; n. By 24.8 the only possibility is the
zero map. Hence A is a _rst order operator. By the general Utiyama theorem,
f factorizes through a map g : g _2Rm_ ! g _3Rm_. The equivariance
of g with respect to the homotheties in G1
m yields a homogeneity condition
c3g(y) = g(c2y), y 2 g _2Rm_. Since there is no integer satisfying 3 = 2d, g is
the zero map. _
The curvature of is a section CP : BP ! LP _2T_BP. According to
the general theory, induces a linear connection ~
on the adjoint bundle LP.
Hence we can construct the covariant exterior di_erential
(2) r~
CP : BP ! LP _3T_BP:
By the geometric character of this construction, (2) determines a gauge natural
operator. Then our lemma implies
(3) r~
CP () = 0:
By 11.15, this is the Bianchi identity for .
53. Base extending gauge natural operators
53.1. Analogously to 18.17, we now formulate the concept of gauge natural
operators in more general situation. Let F, E and H be three G-natural bundles
over m-manifolds.
De_nition. A gauge natural operator D: F (E;H) is a system of regular
operators DP : C1FP ! C1
BP (EP;HP) for every PBm(G)-object P satisfying
D_ P (Ff _ s _ Bf1) = Hf _ DP (s) _ Ef1 for every s 2 C1FP and every
PBm(G)-isomorphism f : P ! _ P, as well as a localization condition analogous
to 51.14.(b).
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
406 Chapter XII. Gauge natural bundles and operators
53.2. Quite similarly to 18.19, one deduces
Proposition. k-th order gauge natural operators F (E;H) are in a canonical
bijection with the natural transformations JrF _ E ! H.
If we have a natural transformation q : H ! E such that every qP : HP ! EP
is a surjective submersion and we require every DP (s) to be a section of qP , we
write D: F (H ! E). Then we _nd in the same way as in 51.15 that the
G-natural operators F (H ! E) are in bijection with the Wsm
-equivariant
maps f : Jk
0 F _ E0 ! H0, satisfying q0 _ f = pr2, where q0 : H0 ! E0 is the
restriction of qRm_G and s is the maximum of the orders in question.
53.3. Gauge natural operators Q (QT ! TB). In 46.3 we deduced that
every connection on principal bundle P ! M with structure group G induces
a connection T on the principal bundle TP ! TM with structure group TG.
Hence T is a (_rst-order) G-natural operator Q (QT ! TB). Now we are
going to determine all _rst-order G-natural operators Q (QT ! TB). Since
the di_erence of two connections on TP ! TBP is a section of L(TP)T_TBP,
it su_ces to determine all _rst-order G-natural operators Q (LT T_TB !
TB). The _ber of the total projection L(T(Rm _ G)) T_TRm ! TRm ! Rm
over 0 2 Rm is the product of Rm with tg T_
0 TRm, 0 2 TRm = R2m. By 53.2
our operators are in bijection with the W2m
G-equivariant maps J1
0Q(Rm _G)_
Rm ! Rm _ tg T_
0 TRm over the identity of Rm.
We know from 10.17 that TG coincides with the semidirect product G o g
with the following multiplication
(1) (g1;X1)(g2;X2) = (g1g2; Ad(g1
2 )(X1) + X2)
where Ad means the adjoint action of G. This identi_es the Lie algebra tg of TG
with g _ g and a direct calculation yields the following formula for the adjoint
action AdTG of TG
(2) AdTG(g;X)(Y; V ) = (Ad(g)(Y ); Ad(g)([X; Y ] + V )):
Hence the subspace 0_g _ tg is AdTG-invariant, so that it de_nes a subbundle
K(TP) _ L(TP). The injection V 7! (0; V ) induces a map IP : LP ! K(TP).
Every modi_ed curvature C(z)P () of a connection on P, see 52.5, can be
interpreted as a linear morphism _2TBP ! LP. Then we can de_ne a linear
map _(C(z)P ()): TTBP ! L(TP) by
(3) _(C(z)P ())(A) = IP (C(z)P ()(_1A ^ _2A)); A 2 TTBP
where _1 : TTBP ! TBP is the bundle projection and _2 : TTBP ! TBP is
the tangent map of the bundle projection TBP ! BP. This determines one
series _(C(z)), z 2 Z, of G-natural operators Q (LT T_TB ! TB).
Moreover, if we consider a modi_ed curvature C(z)P () as a map C(z) : P _
_2TBP ! g, we can construct its vertical prolongation with respect to the _rst
factor
V1C(z)P (): V P _ _2TBP ! Tg = tg:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
53. Base-extending gauge natural operators 407
Then we add the vertical projection _ : TP ! V P of the connection and we
use the projections _1and _2 from (3). This yields a map
(4) _ (C(z)P ()) : TP _ TTBP ! tg
_ (C(z)P ())(U;A) = V1C(z)P ()(_U; _1A ^ _2A); U 2 TP; A 2 TTBP:
The latter map can be interpreted as a section of L(TP)T_TBP, which gives
another series _ (C(z)), z 2 Z, of G-natural operators Q (LT T_TB ! TB).
Proposition. All _rst-order gauge natural operators Q (QT ! TB) form
the following 2dimZ-parameter family
(5) T + _(C(z)) + _ (C(_z)); z; _z 2 Z:
The proof will occupy the rest of this section.
53.4. Let be a connection on Rm _ G with equations
(1) !p = p
i (x)dxi:
Let ("p) be the second component of the Maurer-Cartan form of TG (the _rst
one is (!p)) and let Xi be the induced coordinates on T0Rm. Applying the
description of the Maurer-Cartan form of TG from 37.16 to (1), we _nd the
equation of T is of the form (1) and
(2) "p = @p
i
@xj Xjdxi + p
i dXi:
53.5. Remark _rst that every d
dt
__
0 x(t) 2 T0Rm de_nes a map
(1) T1m
G ! TG; j1
0' 7! d
dt
__
0 (' _ x)(t):
Consider an isomorphism _x = f(x), _y = '(x) _ y of Rm _ G and an element of
V (T(Rm _ G) ! TRm). Clearly, such an element can be generated by a map
(x(t); y(t; u)) : R2 ! Rm _ G, t, u 2 R. This map is transformed into
(2) _x = f(x(t)); _y = '(x(t)) _ y(t; u):
Di_erentiating with respect to t, we _nd
(3) d_y
dt
= T_(d'(x(0))
dt
;
dy(0; u)
dt
)
where _: G _ G ! G is the group composition. This implies that the next
di_erentiation with respect to u yields the adjoint action of TG with respect to
(1). Thus, if (Y p; V p) are the coordinates in tg given by our basis in g, then
we deduce by the latter observation that the action of W2m
G on Rm _ tg is
_X
i = ai
jXj and
(4) _ Y p = Apq
(a)Y q; _ V p = Apq
(a)(cq
rsarj
XjY s + V q):
On the other hand, the action of W2m
G on T0TRm goes through the projection
into G2
m and has the standard form
(5) d_xi = ai
jdxj ; d_X i = ai
jkXjdxk + ai
jdXj :
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
408 Chapter XII. Gauge natural bundles and operators
53.6. Our problem is to _nd all W2m
G-equivariant maps f : Rm _J1
0Q ! Rm _
tg T_
0 TRm over idRm. On J1
0Q, we replace p
ij by Rp
ij and Sp
ij as in 52.3. The
coordinates on tg T_
0 TRm are given by
Y p = Bp
i dxi + Cp
(1) i dXi
V p = Dp
i dxi + Ep
i (2) dXi:
Hence all components of f are smooth functions of X = (Xi), = (p
i ), R =
(Rp
ij ), S = (Sp
ij ). Using 53.5.(4){(5), we deduce from (1) the transformation
laws
_ Cp
i = Apq
(a)Cq
j ~aj
i (3)
_B
p
i = Apq
(a)Bq
j ~aj
i
Apq
(a)Cq
k~akj
aj
li(4) Xl:
Let us start with the component Cp
i (X; ; R; S) of f. Using ap
ij and ap
i , we
deduce that C's are independent of and S. Then we have the situation of the
following lemma.
Lemma. All AdG _ GL(m;R)-equivariant maps Rm _ g _2Rm_ ! g Rm_
have the form _pq
Rq
ijXj with (_pq
) 2 Z.
Proof. First we determine all GL(m;R)-maps h: Rm _ _n 2 Rm_ ! _nRm_,
h = (hp
i (bq
jk;Xl)). If we consider the contraction hh; vi of h with v = (vi) 2 Rm,
we can apply the tensor evaluation theorem to each component of hh; vi. This
yields
hp
i vi = 'p(bq
ijXiXj ; br
ijviXj ; bs
ijXivj ; bt
ijvivj):
Di_erentiating with respect to vi and setting vi = 0, we obtain
(5) hp
i = 'pq
(br
klXkXl)bq
ijXj + p
q (br
klXkXl)bq
jiXj
with arbitrary smooth functions 'pq
, p
q of n variables. If bp
ij = Rp
ij are antisymmetric,
we have Rp
ijXiXj = 0 and Rp
ijXj = Rp
jiXj , so that
(6) hp
i = _pq
Rq
ijXj ; _pq
2 R:
The equivariance with respect to G then yields Apq
(a)_q
r = _pq
Aq
r(a), i.e. (_pq
) 2
Z. _
Thus our lemma implies Cp
i = _pq
Rq
ijXj , (_pq
) 2 Z. For the components Bp
i
of f, the use of ap
i and ap
ij gives that B's are independent of and S. Then the
equivariance with respect to ai
jk yields
(7) Cp
i = 0:
Using our lemma again, we obtain
(8) Bp
i = p
qRq
ijXj ; (p
q ) 2 Z:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 409
53.7. From 53.6.(2) we deduce the transformation laws
_E
p
i = Apq
(a)Eq
j ~aj
i + Apq
(a)cq
rsarj
XjCs
i (1)
_D
p
i = Apq
(a)Dq
j ~aj
i + Apq
(a)cq
rsarj
XjBs
i
_Ep
j aj
(2) kiXk:
By 53.6.(7), the _rst equation implies Ep
i = _pq
Rq
ijXj , (_pq
) 2 Z, in the same way
as in 53.6. Using ap
ij in the second equation, we _nd that the D's are independent
of S. Then the use of ai
jk implies
(3) Ep
i = 0:
The equivariance of D's with a = e, ai
j = _ij
now reads
Dp
i (X; q
j + aq
j ;R) = Dp
i (X; ;R) + cpq
raq
jXjr
sRs
ikXk:
Di_erentiating with respect to aq
j and setting aq
j = 0, we _nd that the D's are
of the form
Dp
i = cpq
rq
jXjr
sRs
ikXk + Fp
i (Xj ;Rq
klXl):
The `absolute terms' Fp
i can be determined by lemma 53.6. This yields
(4) Dp
i = cpq
rq
jXjr
sRs
ikXk + kp
qRq
ijXj ; (kp
q ) 2 Z:
One veri_es easily that (3), (4) together with 53.6.(7){(8) and 53.4.(1){(2) is
the coordinate form of proposition 53.3. _
54. Induced linear connections on the total space of vector and principal bundles
54.1. Gauge natural operators Q _ QTB QT. Given a vector bundle
_ : E ! BE of _ber dimension n, we denote by GL(Rn;E) ! BE the bundle
of all linear frames in the individual _bers of E, see 10.11. This is a principal
bundle with structure group GL(n), n = the _ber dimension of E. Clearly E
is identi_ed with the _ber bundle associated to GL(Rn;E) with standard _ber
Rn. The construction of associated bundles establishes a natural equivalence
between the category PBm(GL(n)) and the category VBm;n := VB \ FMm;n.
A linear connection D on a vector bundle E is usually de_ned as a linear
morphism D: E ! J1E splitting the target jet projection J1E ! E, see section
17. One _nds easily that there is a canonical bijection between the linear
connections on E and the principal connections on GL(Rn;E), see 11.11. That
is why we can say that Q(GL(Rn;E)) =: QE is the bundle of linear connections
on E. In the special case E = TBE this gives a well-known fact from the theory
of classical linear connections on a manifold.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
410 Chapter XII. Gauge natural bundles and operators
An interesting geometrical problem is how a linear connection D on a vector
bundle E and a classical linear connection _ on the base manifold BE can
induce a classical linear connection on the total space E. More precisely, we
are looking for operators which are natural on the category VBm;n. Taking into
account the natural equivalence between VBm;n and PBm(GL(n)), we see that
this is a problem on base-extending GL(n)-natural operators. But we _nd it
more instructive to apply the direct approach in this section. Thus, our problem
is to _nd all operators Q _ QTB QT which are natural on VBm;n.
54.2. First we describe a concrete construction of such an operator. Let us
denote the covariant di_erentiation with respect to a connection by the symbol
of the connection itself. Thus, if X is a vector _eld on BE and s is a section
of E, then DXs is a section of E. Further, let XD denote the horizontal lift
of vector _eld X with respect to D. Moreover, using the translations in the
individual _bers of E, we derive from every section s: BE ! E a vertical vector
_eld sV on E called the vertical lift of s.
Proposition. For every linear connection D on a vector bundle E and every
classical linear connection _ on BE there exists a unique classical linear connection
= (D; _) on the total space E with the following properties
(1)
XDY D = (_XY )D; XDsV = (DXs)V ;
sV XD = 0; sV _V = 0;
for all vector _elds X, Y on BE and all section s, _ of E.
Proof. We use direct evaluation, because we shall need the coordinate expressions
in the sequel. Let xi, yp be some local linear coordinates on E and
Xi = dxi, Y p = dyp be the induced coordinates on TE. If
(2) dyp = Dp
qi(x)yqdxi
are the equations of D and _i(x) @
@xi or sp(x) is the coordinate form of X or s,
respectively, then DXs is expressed by
(3) @sp
@xi _i Dp
qisq_i:
The coordinate expression of XD is
(4) _i @
@xi + Dp
qiyq_i @
@yp
and sV is given by
(5) sp(x) @
@yp :
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 411
Let
(6) dXi = _i
jkXjdxk
be the coordinate expression of _ and let
(7)
dXi = (i
jkXj + i
pkY p)dxk + (i
jqXj + i
pqY p)dyq;
dY p = (p
ijXi + p
qjY q)dxj + (p
irXi + pq
rY q)dyr
be the coordinate expression of . Evaluating (1), we obtain
(8)
i
jk = _i
jk; p
ij =
_
@
@xj Dp
qi
Dp
rjDr
qi + Dp
qk_k
ij
_
yq;
j
ip = j
pi = 0; p
iq = p
qi = Dp
qi; i
pq = 0; pq
r = 0:
This proves the existence and the uniqueness of . _
54.3. Since the di_erence of two classical linear connections on E is a tensor
_eld of TE T_E T_E, we shall heavily use the gauge natural di_erence
tensors in characterizing all gauge natural operators Q _ QTB QT.
The projection T_ : TE ! TBE de_nes the dual inclusion E_T_BE ,! T_E.
The contracted curvature _(D) of D is a tensor _eld of T_BE T_BE. On the
other hand, the Liouville vector _eld L of E is a section of TE. Hence L_(D)
is one of the di_erence tensors we need.
Let _ be the horizontal form of D in the sense of 31.5, so that _ is a tensor
of TE T_E. The contracted torsion tensor ^ S of _ is a section of T_BE and
we construct two kinds of tensor product _ ^ S and ^ S _.
According to 28.13, all natural operators transforming _ into a section of
T_BE T_BE form an 8-parameter family, which we denote by G(_). Hence
L G(_) is an 8-parameter family of gauge natural di_erence tensors. Finally,
let N(_) be the 3-parameter family de_ned in 45.10.
Proposition. All gauge natural operators Q _ QTB QT form the following
15-parameter family
(1)
(1 k1)(D;N(_)) + k1_(D;N(_)) + k2L _(D)+
k3_ ^ S + k4 ^ S _ + L G(_)
where bar denotes the conjugate connection.
We remark that the list (1) is essentially simpli_ed if we assume _ to be
without torsion. Then ^ S vanishes, N(_) is reduced to _ and the 8-parameter
family G(_) is reduced to a two-parameter family generated by the two di_erent
contractions R1 and R2 of the curvature tensor of _. This yields the following
Corollary. All gauge natural operators transforming a linear connection D on
E and a linear symmetric connection _ on TBE into a linear connection on TE
form the following 4-parameter family
(2) (1 k1)(D; _) + k1_(D; _) + L (k2_(D) + k3R1 + k4R2):
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
412 Chapter XII. Gauge natural bundles and operators
54.4. To prove proposition 54.3, _rst we take into account that, analogously to
51.16 and 23.7, every gauge natural operator A: Q _ QTB QT has a _nite
order. Let Sr = Jr
0Q(Rm _ Rn ! Rm) be the _ber over 0 2 Rm of the r-th
jet prolongation of the connection bundle of the vector bundle Rm _ Rn ! Rm,
let Zr = Jr
0TRm be the _ber over 0 2 Rm of the r-th jet prolongation of the
connection bundle of TRm and V be the _ber over 0 2 Rm of the connection
bundle of T(Rm _ Rn) with respect to the total projection QT(Rm _ Rn) !
(Rm _ Rn) ! Rm. Then all Sr, Zr, Rn and V are Wr+1
m (GL(n)) =: Wr+1
m;n -
spaces. In fact, Wr+1
m;n acts on Zr by means of the base homomorphisms into
Gr+1
m , on Rn by means of the canonical projection into GL(n) and on V by means
of the jet homomorphism into W1
m;n. The r-th order gauge natural operators
A: Q _ QTB QT are in bijection with Wr+1
m;n -equivariant maps (denoted by
the same symbol) A: Sr_Zr_Rn ! V satisfying q _A = pr3, where q : V ! Rn
is the canonical projection.
Formula 54.2.(2) induces on Sr the jet coordinates
(1) D_ = (Dp
qi_); 0 _ j_j _ r
where _ is a multi index of range m. On Zr, 54.2.(6) induces analogously the
coordinates
(2) __ = (_i
jk_); 0 _ j_j _ r:
On V , we consider the coordinates y = (yp) and
(3) A
BC; A; B; C = 1; : : : ;m + n
given by 54.2.(7). Hence the coordinate expression of any smooth map f : Sr _
Zr _ Rn ! V satisfying q _ f = pr3 is yp = yp and
(4) A
BC = fA
BC(D_;__; y):
The coordinate form of a linear isomorphism of vector bundle Rm_Rn ! Rm
is
(5) _xi = fi(x); _yp = fp
q (x)yq:
The induced coordinates on Wr+1
m;n are
ai
_ = @_fi(0); ap
q_ = @_fp
q (0); 0 < j_j _ r + 1; 0 _ j_j _ r + 1:
The above-mentioned homomorphism Wr+1
m;n
! Gr+1
m consists in suppressing the
coordinates ap
q_.
The standard action of G2
m on Z0 is given by 25.2.(3). The action of W1
m;n
on S0 is a special case of 52.1.(5) for the group G = GL(n). This yields
(6) _D p
qi = aprr
sj~asq
~aj
i + ap
rj~arq
~aj
i :
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 413
The canonical action of GL(n) on Rn is
(7) _yp = apq
yq:
Using standard evaluation, we deduce from 54.2.(7) that the action of W1
m;n on
V is (7) and
ai
jk + ail
l
jk = _i
lmal
jamk
+_
i
plap
qjyqal
k +_
i
lpal
jap
qkyq +_i
pqap
rjyraq
(8) skys
ail
l
pj = _i
qkaq
pakj
+_
i
qraq
par
(9) sjys
ail
l
jp = _i
kqakj
aq
p +_
i
qraq
sjysar
p (10)
ail
l
pq = _
i
rsar
pasq
(11)
ap
qijyq + ap
qkyqk
ij + apq
q
ij = _
p
klaki
al
j +_
p
qlaq
riyral
j(12) +
_
p
kqaki
aq
rjyr +_pq
raq
siysar
tjyt
ap
rkyrkq
i + ap
qi + apr
rq
i = _
p
rkarq
aki
+_pr
sarq
as
(13) tiyt
ap
rkyrk
iq + ap
qi + apr
r
iq = _
p
jraj
i arq
+_pr
sar
tiytasq
(14)
ap
sjysj
qr + aps
sq
r = _
p
stasq
at
r (15)
54.5. Let H _ Wr+1
m;n be the subgroup determined by the (r + 1)-th jets of the
products of linear isomorphisms on both Rm and Rn, which is canonically isomorphic
to GL(m)_GL(n). The standard prolongation procedure and 54.5.(8){(15)
imply that the actions of H on Dp
qi_, _i
jk_ and A
BC are tensorial.
Consider the equivariance of fi
pq with respect to the _ber homotheties. This
yields
k2fi
pq = fi
pq(D_;__; ky):
Multiplying by k2 and letting k ! 0, we obtain
(1) i
pq = fi
pq = 0:
The equivariance of fi
jp with respect to the _ber homotheties gives
k1fi
jp = fi
jp(D_;__; ky):
This implies in the same way
(2) i
jp = 0:
For fi
pj and fp
qr we _nd quite similarly
(3) i
pj = 0; pq
r = 0:
For fp
qi the _ber homotheties give
fp
qi = fp
qi(D_;__; ky):
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
414 Chapter XII. Gauge natural bundles and operators
Letting k ! 0 we _nd fp
qi are independent of yp. Then the base homotheties
yield
kfp
qi = fp
qi(k1+j_jD_; k1+j_j__):
By the homogeneous function theorem, fp
qi are linear in Dp
qi, _i
jk and independent
of D_, __ with j_j > 0, j_j > 0. By the generalized invariant tensor theorem,
we obtain
(4) fp
qi = aDp
qi + b_p
qDr
ri + c_p
q_j
ji + d_p
q_j
ij :
Let K _ Wr+1
m;n be the subgroup characterized by ai
j = _ij
, apq
= _p
q . By 25.2.(3),
54.4.(6) and 54.4.(13), the equivariance of (4) on K reads
ap
qi = aap
qi + b_p
q arr
i + c_p
q aj
ji + d_p
q ai
ij :
This implies a = 1, b = 0, c + d = 0, i.e.
(5) p
qi = Dp
qi + c1_p
q(_j
ji
_j
ij); c1 2 R:
For fp
iq we deduce in the same way
(6) p
iq = Dp
qi + c2_p
q(_j
ji
_j
ij); c2 2 R:
The _ber homotheties yield that fi
jk is independent of yp. Then the base
homotheties imply that fi
jk is linear in Dp
qi, _i
jk and independent of Dp
qi_, _i
jk_
with j_j > 0, j_j > 0. By the generalized invariant tensor theorem, we obtain
(7)
fi
jk = a_i
jk + b_i
kj + c_ij
_l
lk + d_ij
_l
kl+
e_ik
_l
lj + f_ik
_l
jl + g_ij
Dp
pk + h_ik
Dp
pj:
By 25.2.(3), 54.4.(6) and 54.4.(8), the equivariance of (7) on K reads
ai
jk = (a + b)ai
jk + (c + d)_ij
al
lk + (e + f)_ik
al
lj + g_ij
ap
pk + h_ik
ap
pj:
This implies a + b = 1, c + d = e + f = g = h = 0, i.e.
(8) i
jk = (1 c3)_i
jk + c3_i
kj + c4_ij
(_l
lk
_l
kl) + c5_ik(_l
lj
_l
jl):
54.6. The study of fp
ij is quite analogous to 54.5, but it leads to more extended
evaluations. That is why we do not perform all of them in detail here. The _ber
homotheties yield
kfp
ij = fp
ij(D_;__; ky):
By the homogeneous function theorem, fp
ij is linear in yp, i.e.
fp
ij = Fp
ijq(D_;__)yq:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 415
The base homotheties then imply
k2Fp
ijq = Fp
ijq(k1+j_jD_; k1+j_j__):
By the homogeneous function theorem, Fp
ijq is linear in Dp
qij , _i
jkl, bilinear in
Dp
qi, _i
jk and independent of Dp
qi_, _i
jk_ with j_j > 1, j_j > 1. Using the generalized
invariant tensor theorem, we obtain Fp
ijq in the form of a 40-parameter
family. The equivariance of fp
ij with respect to K then yields
p
ij =
_
(1 c6)Dp
qij + c6Dp
qji + c7_p
q (Dr
rij
Dr
(1) rji)
c6Dp
riDr
qj + (c6 1)Dp
rjDr
qi + (1 c3)Dp
qk_k
ij + c3Dp
qk_kj
i+
(c4 c1)Dp
qi(_l
lj
_l
jl) + (c5 c2)Dp
qj(_l
li
_l
il)+
_p
qGij(_)
_
yq
where Gij(_) is the coordinate form of G(_).
One veri_es directly that (1) and 54.5.(1){(3), (5), (6), (8) is the coordinate
expression of 54.3.(1). _
54.7. The case of principal bundles. An analogous problem is to study the
gauge natural operators transforming a connection D on a principal G-bundle
_ : P ! BP and a classical linear connection _ on the base manifold BP into a
classical linear connection on the total space P. First we present a geometrical
construction of such an operator.
Let vA be the vertical component of a vector A 2 TyP and bA be its projection
to the base manifold. Consider a vector _eld X on BP such that j1
xX = _(bA),
x = _(y). Construct the lift XD of X and the fundamental vector _eld '(vA)
determined by vA. An easy calculation shows that the rule
(1) A 7! j1
y(XD + '(vA))
determines a classical linear connection NP (D; _): TP ! J1(TP ! P) on P.
54.8. We are going to determine all gauge natural operators of the above type.
The result of 54.3 suggests us that the case _ is without torsion is much simpler
than the general case. That is why we restrict ourselves to a symmetric _. Since
the di_erence of two classical linear connections on P is a tensor _eld of type
TP T_P T_P, we characterize all gauge natural operators in question as
a sum of the operator N from 54.7 and of the gauge natural di_erence tensor
_elds. We construct geometrically the following 3 systems of di_erence tensor
_elds.
I. The connection form of D is a linear map !: TP ! g. Take any bilinear
map f1 : g _ g ! g and compose ! _ ! with f1. This de_nes an n3-parameter
system of di_erence tensor _elds TP TP ! V P, n = dimG.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
416 Chapter XII. Gauge natural bundles and operators
II. The curvature form D! of ! is a bilinear map TP _ TP ! g. Take any
linear map f2 : g ! g and compose D! with f2. This yields an n2-parameter
system of di_erence tensor _elds.
III. By 28.7, all natural operators transforming a linear symmetric connection
_ on BP into a tensor _eld of T_BP T_BP form a 2-parameter family linearly
generated by both di_erent contractions R1 and R2 of the curvature tensor of _.
The tangent map of the bundle projection P ! BP de_nes the dual injection
P _ T_BP ! T_P. Taking any fundamental vector _eld _ Y determined by a
vector Y 2 g, we obtain a 2n-parameter system of di_erence tensor _elds linearly
generated by _ Y R1 and _ Y R2.
54.9. Proposition. All gauge natural operators transforming a connection on
P and a classical linear symmetric connection of the base manifold BP into
a classical linear connection on P form the (n3 + n2 + 2n)-parameter family
generated by operator N and by the above families I, II, and III of the di_erence
tensor _elds.
The proof consists in straightforward application of our techniques, but it is
too long to be performed here. We refer the reader to [Kol_a_r, to appear a].
Remarks
Our approach to gauge natural bundles and operators generalizes directly the
theory of natural bundles. So we also prove the regularity originally assumed
in [Eck, 81]. Let us mention that, analogously to chapter XI, we can de_ne
the Lie derivative of sections of gauge natural bundles with respect to the right
invariant vector _elds on the corresponding principal _ber bundles and then
the in_nitesimally gauge natural operators. The relation between the gauge
naturality and in_nitesimal gauge naturality is similar to the case of natural
bundles if the gauge group is connected; more information can be found in [Cap,
Slov_ak, 92].
The _rst application of our methods for _nding gauge natural operators was
presented in [Kol_a_r, 87b]. The considerations in that paper are restricted to
the case the structure group is the general linear group GL(n) in an arbitrary
dimension (independent of the dimension of the base manifold), for in such a
case one can apply directly the results from chapter VI. [Kol_a_r, 87b] has also
determined all GL(n)-natural operators transforming a principal connection on
a principal bundle P and a classical linear connection on the base manifold into
a principal connection on W1P. The curvature-like operators were found in the
special case G = GL(n) in [Kol_a_r, 87b] and the general problem was solved
in [Kol_a_r, to appear a]. The greater part of the results from section 52 was
deduced in [Kol_a_r, to appear b]. Proposition 53.3 was proved for the special
case G = GL(n) in [Kol_a_r, 91], the general result is _rst presented in this book.
Section 54 is based on [Gancarzewicz, Kol_a_r, 91].
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
417
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Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
428 List of symbols
List of symbols
1j the multi index with j-th component one and all others zero, 13.2
_: Jr(M;N) ! M the source mapping of jets, 12.2
_ : Jr(M;N) ! N the target mapping of jets, 12.2
B: FM!Mf the base functor, 2.20
C1E, also C1(E ! M) the space of smooth sections of a _ber bundle
C1(M;N) the space of smooth maps of M into N
C1
x (M;N) the space of germs at x 2 M, 1.4
conja : G ! G the conjugation in a Lie group G by a 2 G, 4.24
d usually the exterior derivative, 7.8
D the algebra of dual numbers, 37.1
Dr
n = Jr
0 (Rn;R) the algebra of r-jets of functions, 40.5
(E; p;M; S), also simply E usually a _ber bundle with total space E, base M,
and standard _ber S, 9.1
F usually the ow operator of a natural bundle F, 6.19, 42.1
FlXt
, also Fl(t;X) the ow of a vector _eld X, 3.7
FM the category of _bered manifolds and _ber respecting mappings, 2.20
FMm the category of _bered manifolds with m-dimensional bases and _ber
respecting mappings with local di_eomorphisms as base maps, 12.16
FMm;n the category of _bered manifolds with m-dimensional bases and ndimensional
_bers and locally invertible _ber respecting mappings,
17.1
FM_ the category of star bundles, 41.1
G usually a general Lie group with multiplication _ : G _ G ! G, left
translation _, and right translation _
g the Lie algebra of a Lie group G
Gr
m the jet group (di_erential group) of order r in dimension m, 12.6
Gr
m;n the jet group of order r of the category FMm;n, 18.8
GL(n) the general linear group in dimension n with real coe_cients, 4.30
GL(Rn;E) the linear frame bundle of a vector bundle E, 10.11
Ik short for the k _ k-identity matrix IdRk
invJr(M;N) the bundle of invertible r-jets of M into N, 12.3
JrE the bundle of r-jets of local sections of a _ber bundle E ! M, 12.16
Jr(M;N) the bundle of r-jets of smooth functions from M to N, 12.2
jrf(x), also jrx
f the r-jet of a mapping f at x, 12.2
Krn
the functor of (n; r)-contact elements, 12.15
L the Lie derivative, 6.15, 47.4
` : G _ S ! S usually a left action of a Lie group
L(V;W) the space of all linear maps of vector space V into a vector space W
LP = P[g; Ad] the adjoint bundle of principal bundle P(M;G), 17.6
Lr the r-th order skeleton of the category Mf, 12.6
M usually a (base) manifold
Mf the category of manifolds and smooth mappings, 1.2
Mfm the category of m-dimensional manifolds and local di_eomorphisms,
6.14
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Author index 429
N natural numbers
k(M) the space of k-forms on a manifold M, 7.4
k(M;E) the space of E-valued k-forms, 7.11
P(M;G), also (P; p;M;G) a principal _ber bundle with structure group G,
10.6
P[S; `], also P[S] the associated bundle to a principal bundle P(M;G) for the
action ` : G _ S ! S, 10.7
PB the category of principal _ber bundles, 10.6
PBm the category of principal bundles over m-dimensional manifolds and of
PB-morphisms covering local di_eomorphisms, 17.4
PB(G) the category of principal G-bundles, 10.6
PBm(G) the category of principal G-bundles over m-dimensional manifolds
and local isomorphisms, 15.1
PrM = invJr
0 (RdimM;M) the r-th order frame bundle of a manifold M, 12.12
_r
s : Jr(M;N) ! Js(M;N) projection of r-jets into s-jets, s _ r, 12.2
QP the connection bundle of a principal bundle P, 17.4
Q_P1M the bundle of torsion free linear connections, 25.3
R real numbers
r : P _ G ! P usually a right action, in particular the principal right action of
a principal bundle
TM the tangent bundle of a manifold M with projection _M : TM ! M
1.7
T(r)M = Jr(M;R)_
0 the r-order tangent bundle, 12.14
Tr
k = Jr
0 (Rk; ) the functor of (k; r)-velocities, 12.8
TA the Weil functor corresponding to the Weil algebra A, 35.11
tx usually the translation Rm ! Rm, y 7! y + x
VB the category of vector bundles, 6.3
Wrm
G the (m; r)-principal prolongation of a Lie group G, 15.2
WrP the r-th principal prolongation of a principal bundle P, 15.3
X(M) the set of all vector _elds on a manifold M, 3.1
Y ! M usually a _bered manifold
Z integers
Author index
Albert, 48
Atiyah, 265, 266, 267, 295
Baston, 281, 293, 294, 295
Bernstein, 292
Boe, 294
Boerner, 131, 281
Boman, 172
Bott, 265, 266, 267, 295
Branson, 293, 295
Brocker, 10
Cahen, 210
Cap, 252, 254, 393, 416
Carrell, 133, 216, 218
Chrastina, 181, 210
Cohen, 11
Collingwood, 294
de Le_on, 356
De Wilde, 210, 252
Dekr_et, 257
Dieudonn_e, 9, 17, 133, 216, 218
Donaldson, 5
Doupovec, 229, 359, 375
Eastwood, 281, 293, 294
Eck, 297, 328, 400, 416
430 Author index
Ehresmann, 117, 166, 167, 169, 265
Eilenberg, 169
Epstein, 116, 168, 188, 210, 295
Fegan, 294
Feigin, 292
Feng Luo, 5
Freedman, 4, 5
Frolicher, 59, 75, 79, 396
Fuks, 292
Gancarzewicz, 345, 357, 416
Gheorghiev, 167
Gilkey, 275, 295
Goldschmidt, 340
Gompf, 5
Graham, 295
Greub, 5, 81, 115
Grozman, 291
Gurevich, 215
Gutt, 210
Halperin, 5, 81, 115
Hilgert, 48
Hirsch, 10, 11, 82, 180, 314, 330
Hochschild, 201
Jacobson, 42
Jany_ska, 166, 248, 357, 393
Joris, 12
Janich, 10
Kainz, 297, 328
Kirillov, 281, 282, 289, 292, 393
Kobak, 363
Kobayashi, 100, 107, 162, 166, 167, 403
Kock, 349
Kowalski, 277, 295
Kriegl, 59, 79, 310
Krupka, 248, 252, 357, 360
Kurek, 232, 265, 402
Laptev, 167
Lecomte, 48, 252, 404
Leites, 292
Libermann, 115, 167
Lichnerowicz, 235, 243
Lubczonok, 248
Luciano, 297, 328
Luna, 224
Malgrange, 178
Mangiarotti, 75, 340
Mauhart, 48, 60
Mikol_a_sov_a, 252
Mikulski, 210, 211, 343, 349, 360
Milnor, 4, 301
Modugno, 227, 257, 262, 263, 295, 340, 393
Molino, 48
Montgomery, 43, 45, 310
Morimoto, 297, 355
Morrow, 81
Nagata, 5, 81
Naymark, 130, 131, 285
Neeb, 48
Newlander, 75
Nijenhuis, 68, 75, 116, 138, 210, 255
Nirenberg, 75
Nomizu, 81, 100, 107, 162, 166, 403
Ozeki, 81
Palais, 83, 116, 138, 168, 210, 222, 282
Patodi, 265, 266, 267, 295
Peetre, 176, 210
Pohl, 363
Pradines, 334
Quinn, 4
Radziszewski, 229, 248
Reinhart, 166
Rice, 294
Richardson, 68
Rodriguez, 356
Rudakov, 286, 288
Sattinger, 48
Saunders, 393
Schouten, 248, 255
Sekizava, 277, 295, 357
Shmelev, 292
Shtern, 285
Stashe_, 301
Stefan, 48
Stefani, 227
Sternberg, 340
Stredder, 276, 295
Sussman, 48
Terng, 116, 128, 136, 138, 166, 168, 210, 282,
286
Thurston, 116, 168, 188, 210,
Tougeron, 178
Trautman, 392
Tulczyjew, 227
van Strien, 252
Vanstone, 5, 81, 115
Varadarajan, 42
Virsik, 167
Vosmansk_a, 265, 295, 349
Weaver, 48
Weil, 296, 301, 328
Weyl, 265
White, 393
Whitney, 10, 178
Wolf, 115
Yamabe, 43
Zajtz, 210, 375
Zhelobenko, 285
Zippin, 43, 45, 310
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Index 431
Index
A
A-admissible, 341
A-velocity, 306
absolute di_erential, 164
absolute di_erential along X, 163, 164
absolute operator, 351
action of a category, 147
adjoint representation, 38
admissible category, 171
a_ne bundle, 60
a_ne bundle functor on Mfm, 142
algebraic bracket, 68
algebraic description of Weil functors, 305
almost complex structure, 75
almost Whitney-extendible, 184
anholonomic, 16
associated bundle, 90
associated map, 171, 174
associated map to the k-th order operator A,
143
associated maps of the bundle functor F, 139
atlas, 4
B
base, 11, 50
base extending, 173
base functor B: FM!Mf, 15
Bianchi identity, 78
Borel subalgebra, 285
bundle functor on Mfm, 138
bundle functor on C, 170
bundle functor on the category Mf, 146
C
C-connection, 365
C-_eld, 175, 365
Cr-map, 173
canonical ip mapping, 55, 319
canonical form on P1M, 155
canonical lift, 59
Cartan subalgebra, 285
category over manifolds, 169
_Cech cohomology set, 51
center, 44
centralizer, 44
chart, 4
chart description of Weil functors, 301
Chern forms, 269
Christo_el forms, 79
classical complex Lie groups, 32
classifying spaces, 94
closed form, 66
cocurvature, 73
cocycle condition, 51, 77
cocycle of transition functions, 51, 77
cohomologous, 51, 87
cohomology classes, 51
complete connection, 81
complete vector _eld, 19
complete ow, 19
completely reducible, 131
conformal, 271
conformal weight, 293
conjugation, 38
connection, 73, 77
connection form, Lie algebra valued, 100
connection morphism, 364
connection, general, 77, 158
connector, 110, 326
contact (n; r)-element, 124
cotangent bundle, 61
covariant derivative, 110, 326
covariant exterior derivative, 103, 111
covelocities, 120
curvature, 73, 111
curvature form, Lie algebra-valued, 100
D
derivation, 6, 322
derivation, graded, 67
derived group, 130
descending central sequence, 130
di_eomorphic, 5
di_eomorphism, 5
di_erential, 8
di_erential form, 62
di_erential group, 119
distinguished chart, 27
distribution, 24
divergence, 131
dual natural vector bundles, 142
dual numbers, 318
E
E-valued k-form, 67, 403
e_ective action, 44
Ehresmann connection, 81
elementary invariant, 265
elementary invariant tensors of degree r, 214
embedding, 9
Euler morphism, 387
evolution operator, 29
exact form, 66
expansion, 321
expansion property, 321
exponential mapping, 36
432 Index
exterior derivative, 65
exterior form, 62
F
f-dependent, 73
f-related, 19, 73
F-metric, 278
_ber, 50
_ber bundle, 76
_ber bundle atlas, 77
_ber chart, 77
_ber over x 2 N, 15
_ber r-jet at x, 395
_bered manifold, 11
_bered product, 15
_nite order r, 139
_rst polarization, 219
ow line, 17
ow prolongation, 59
ow-natural transformation, 336, 338
ow of a vector _eld, 18
foliation, 25
formal curvature map, 234
formally real algebra, 297
frame bundle, 122
frame _eld, 16, 52
frame form, 95, 154, 156
free action, 44
Frolicher-Nijenhuis bracket, 69
fundamental vector _eld, 46
G
G-atlas, 86
G-bundle, 86
G-bundle structure, 86
G-equivariant, 47
G-module, 131
g-module homomorphism, 131
gauge natural bundle, 395
gauge natural operator, 398
gauge transformations, 95
general connection, 77, 158
general linear group, 30
general Ricci identity, 235
generalized Chern-Weil forms, 403
generalized covariant derivative, 378
generalized covariant di_erential, 378
generalized invariant tensor, 230
generalized invariant tensor theorem, 230
generalized Lie derivative, 376
germ of f at x, 6
global ow, 19
grading, 128
Grassmann manifold, 88
H
higher order connections, 160
highest weight vector, 285
holonomic frame _eld, 16
holonomous, 61
holonomy group, 82, 106
holonomy Lie algebra, 82
homogeneous degree, 131
homogeneous function theorem, 213
homogeneous in the order, 284
homogeneous space, 45
homomorphism of G-bundles, 92
homomorphism over _ of principal bundles, 89
horizontal bundle, 78
horizontal di_erential forms, 103
horizontal foliation, 79
horizontal lift, 78, 278
horizontal projection, 78
horizontal space, 73
horizontal vector, 78
I
I-equivalent, 306
ideal, 43
idealisator of a module, 298
idealizer, 44
immersed submanifold, 12
immersion, 11
induced action, 172
induced connection, 107, 108
in_nite jet, 125
in_nite jet prolongation J1Y of Y , 126
in_nitesimal automorphism, 26
in_nitesimally admissible, 170
in_nitesimally regular, 365
initial submanifold, 12
inner automorphism, 38
insertion operator, 63
integrable distribution, 25
integral curve, 17
integral manifold, 24
invariant tensor of degree r, 214
invariant tensor theorem, 214
invariant subspace, 131
involution, 55
involutive distribution, 28
irreducible principle connection, 107
irreducible representation, 131
J
jet, 117
jet group, 119
jet of order 1, 125
jet prolongation, 117
K
k-admissible domain, 176
k-form, 62
Index 433
L
Lagrangian, 387
leaf, 25
left action, 44
left invariant, 33
left logarithmic derivative, 39
Lie algebra, 17
Lie bracket of vector _elds, 17, 325
Lie derivative, 20, 57, 63, 69
Lie group, 30
Lie subgroup, 41
linear connection, 109, 110
linear frame bundle, 94
linear vector _eld, 379
Liouville vector _eld, 257
local category, 169
local di_eomorphism, 5
local isomorphism, 170
local operator, 143
local order, 205
local skeleton, 170
local trivialization, 77
local vector _eld, 16
localization of A over U, 169
locally at category, 185
locally homogeneous, 170
locally non-constant, 179
M
Maurer-Cartan form, 39, 79
maximal integral manifold, 24
method of di_erential equations, 245
mixed curvature, 232
modi_ed curvature operator, 402
morphism operator, 173
morphism of _bered manifolds, 15
multihomogeneous component, 218
multiindex, 118
multilinear version of Peetre theorem, 179
multipolarization, 219
N
natural a_ne bundle, 142
natural bilinear concomitants, 75
natural bundle, 138
natural operator, 143, 174
natural transformation, 58
natural vector bundle, 56, 141
Nijenhuis tensor, 75
Nijenhuis-Richardson bracket, 68
nilpotent, 130
nonlinear Peetre theorem, 179
normalizer, 44
O
object of type _, 170
one parameter subgroup, 35
orbit, 44
order (s; r), 401
orthogonal group, 31
orthonormal frame bundle, 94
P
plaque, 27
point property, 330, 335
pointed local skeleton, 170
polarization technique, 218
polynomial map, 218
Pontryagin forms, 270
principal bundle atlas, 87
principal connection, 100, 159
principal _ber bundle, 87
principal _ber bundle homomorphism, 89
principal prolongations of Lie groups, 150
principal prolongations of principal bundles,
150
principal right action, 87
product of manifolds, 10
product preserving functor, 308
pseudo tensorial forms of type `, 154
pullback, 78
pullback vector bundle, 53
pure manifold, 4
Q
quaternionic unitary group, 33
quaternionically linear, 33
quaternionically unitary, 33
R
r-th jet group (or the r-th di_erential group)
of type _, 171
r-th order curvature equations, 235
r-th order curvature subspace, 236
r-th order frame bundle of A, 171
r-th order Ricci equations, 240
r-th order Ricci subspace, 240
r-th order skeleton of C, 171
r-th order tangent vector, 123
reduction of the structure group, 90
regular functor, 171
regular operator, 143
representation, 38, 131
restricted holonomy group, 82, 106
restricted Lie derivative, 377
Riemannian metric , 94
right action, 44
right invariant, 34
right logarithmic derivative, 38
roots, 285
S
Sasaki lift, 278
second order di_erential equation, 257
434 Index
second semiholonomic prolongation of Y , 262
section, 50
semidirect product, 48
semidirect product of an algebra and a
module, 298
singular foliation, 25
singular vector, 283
smooth distribution, 24
smooth functor, 53
smooth partition of unity, 5
smooth splitting property, 171
solvable, 130
source, 117
special linear group, 31
special orthogonal group, 31
special unitary group, 32
sphere, 9
spray, 257
stable distribution, 26
standard _ber, 49, 77, 172
standard _ber of the bundle functor F, 139
star bundle functor, 346
stereographic atlas, 9
Stiefel manifold, 89
Study numbers, 318
submanifold, 9
submanifold chart, 9
submersion, 11
support, 5, 50, 176
support of a vector _eld, 19
symplectic group, 31
system of standard _bers of the bundle
functor, 147
T
tangent bundle, 8
tangent space of M at x, 7
tangent vector, 6
target, 117
tensor evaluation theorem, 224
tensor _eld, 59, 61
tensorial forms of type `, 154
time dependent vector _eld, 29
topological manifold, 4
torsion form, 155
torsion tensor, 155
torsion-free, 155
total polarization of f, 219
total space, 11, 50, 77
transgression, 270
transition function, 49, 77
transitive action, 44
transversal mapping, 14
typical _ber, 49
U
underlying manifold, 169
unitary group, 32
universal vector bundle, 97
V
vector bundle, 50
vector bundle atlas, 50
vector bundle chart, 49
vector bundle functor on Mfm, 141
vector bundle functor, 56
vector bundle homomorphism, 50
vector bundle isomorphism, 50
vector _eld, 16
vector _eld __ dual to a linear vector _eld _,
380
vector _eld along f, 376
vector sub bundle, 52
vector valued di_erential form, 68
velocities of order r and dimension m, 120
vertical bundle, 55, 77, 98
vertical lift, 55, 278, 319
vertical projection, 55, 78
vertical prolongation, 255
vertical space, 73
vertical Weil bundle, 336
W
W-extendible, 178, 205
weakly local functor, 313
weight, 271, 285
Weil algebra, 298
Whitney extension theorem, 178
Whitney-extendible, 178
width of a Weil algebra, 299
Z
zero section, 50
zero set, 5
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993__