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C-¬eld, 175, 365 distribution, 24
C r -map, 173 divergence, 131
canonical ¬‚ip mapping, 55, 319 dual natural vector bundles, 142
canonical form on P 1 M , 155 dual numbers, 318
canonical lift, 59
E
Cartan subalgebra, 285
category over manifolds, 169 E-valued k-form, 67, 403
ˇ
Cech cohomology set, 51 e¬ective action, 44
center, 44 Ehresmann connection, 81
centralizer, 44 elementary invariant, 265
chart, 4 elementary invariant tensors of degree r, 214
chart description of Weil functors, 301 embedding, 9
Chern forms, 269 Euler morphism, 387
Christo¬el forms, 79 evolution operator, 29
classical complex Lie groups, 32 exact form, 66
classifying spaces, 94 expansion, 321
closed form, 66 expansion property, 321
cocurvature, 73 exponential mapping, 36
432 Index


exterior derivative, 65 highest weight vector, 285
exterior form, 62 holonomic frame ¬eld, 16
holonomous, 61
F holonomy group, 82, 106
f -dependent, 73 holonomy Lie algebra, 82
f -related, 19, 73 homogeneous degree, 131
F-metric, 278
homogeneous function theorem, 213
¬ber, 50
homogeneous in the order, 284
¬ber bundle, 76
homogeneous space, 45
¬ber bundle atlas, 77
homomorphism of G-bundles, 92
¬ber chart, 77
homomorphism over ¦ of principal bundles, 89
¬ber over x ∈ N , 15
horizontal bundle, 78
¬ber r-jet at x, 395
horizontal di¬erential forms, 103
¬bered manifold, 11
horizontal foliation, 79
¬bered product, 15
horizontal lift, 78, 278
¬nite order r, 139
horizontal projection, 78
¬rst polarization, 219
horizontal space, 73
¬‚ow line, 17
horizontal vector, 78
¬‚ow prolongation, 59
¬‚ow-natural transformation, 336, 338 I
¬‚ow of a vector ¬eld, 18 I-equivalent, 306
foliation, 25 ideal, 43
formal curvature map, 234
idealisator of a module, 298
formally real algebra, 297
idealizer, 44
frame bundle, 122
immersed submanifold, 12
frame ¬eld, 16, 52
immersion, 11
frame form, 95, 154, 156
induced action, 172
free action, 44
induced connection, 107, 108
Fr¨licher-Nijenhuis bracket, 69
o
in¬nite jet, 125
fundamental vector ¬eld, 46
in¬nite jet prolongation J ∞ Y of Y , 126
in¬nitesimal automorphism, 26
G
in¬nitesimally admissible, 170
G-atlas, 86
in¬nitesimally regular, 365
G-bundle, 86
initial submanifold, 12
G-bundle structure, 86
inner automorphism, 38
G-equivariant, 47
insertion operator, 63
G-module, 131
g-module homomorphism, 131 integrable distribution, 25
integral curve, 17
gauge natural bundle, 395
integral manifold, 24
gauge natural operator, 398
gauge transformations, 95 invariant tensor of degree r, 214
general connection, 77, 158 invariant tensor theorem, 214
general linear group, 30 invariant subspace, 131
general Ricci identity, 235 involution, 55
generalized Chern-Weil forms, 403 involutive distribution, 28
generalized covariant derivative, 378 irreducible principle connection, 107
generalized covariant di¬erential, 378 irreducible representation, 131
generalized invariant tensor, 230
J
generalized invariant tensor theorem, 230
jet, 117
generalized Lie derivative, 376
jet group, 119
germ of f at x, 6
jet of order ∞, 125
global ¬‚ow, 19
jet prolongation, 117
grading, 128
Grassmann manifold, 88
K
H k-admissible domain, 176
higher order connections, 160 k-form, 62
Index 433


L orbit, 44
Lagrangian, 387 order (s, r), 401
leaf, 25 orthogonal group, 31
left action, 44 orthonormal frame bundle, 94
left invariant, 33
P
left logarithmic derivative, 39
plaque, 27
Lie algebra, 17
point property, 330, 335
Lie bracket of vector ¬elds, 17, 325
pointed local skeleton, 170
Lie derivative, 20, 57, 63, 69
polarization technique, 218
Lie group, 30
polynomial map, 218
Lie subgroup, 41
Pontryagin forms, 270
linear connection, 109, 110
principal bundle atlas, 87
linear frame bundle, 94
principal connection, 100, 159
linear vector ¬eld, 379
principal ¬ber bundle, 87
Liouville vector ¬eld, 257
principal ¬ber bundle homomorphism, 89
local category, 169
principal prolongations of Lie groups, 150
local di¬eomorphism, 5
principal prolongations of principal bundles,
local isomorphism, 170
150
local operator, 143
principal right action, 87
local order, 205
product of manifolds, 10
local skeleton, 170
product preserving functor, 308
local trivialization, 77
pseudo tensorial forms of type , 154
local vector ¬eld, 16
pullback, 78
localization of A over U , 169
pullback vector bundle, 53
locally ¬‚at category, 185
pure manifold, 4
locally homogeneous, 170
locally non-constant, 179
Q
quaternionic unitary group, 33
M
quaternionically linear, 33
Maurer-Cartan form, 39, 79
quaternionically unitary, 33
maximal integral manifold, 24
method of di¬erential equations, 245
R
mixed curvature, 232
r-th jet group (or the r-th di¬erential group)
modi¬ed curvature operator, 402
of type ±, 171
morphism operator, 173
r-th order curvature equations, 235
morphism of ¬bered manifolds, 15
r-th order curvature subspace, 236
multihomogeneous component, 218
r-th order frame bundle of A, 171
multiindex, 118
r-th order Ricci equations, 240
multilinear version of Peetre theorem, 179
r-th order Ricci subspace, 240
multipolarization, 219
r-th order skeleton of C, 171
N r-th order tangent vector, 123
reduction of the structure group, 90
natural a¬ne bundle, 142
regular functor, 171
natural bilinear concomitants, 75
regular operator, 143
natural bundle, 138
representation, 38, 131
natural operator, 143, 174
restricted holonomy group, 82, 106
natural transformation, 58
restricted Lie derivative, 377
natural vector bundle, 56, 141
Riemannian metric , 94
Nijenhuis tensor, 75
right action, 44
Nijenhuis-Richardson bracket, 68
right invariant, 34
nilpotent, 130
right logarithmic derivative, 38
nonlinear Peetre theorem, 179
roots, 285
normalizer, 44

S
O
Sasaki lift, 278
object of type ±, 170
one parameter subgroup, 35 second order di¬erential equation, 257
434 Index


second semiholonomic prolongation of Y , 262 total polarization of f , 219
section, 50 total space, 11, 50, 77
semidirect product, 48 transgression, 270
semidirect product of an algebra and a transition function, 49, 77
module, 298 transitive action, 44
singular foliation, 25 transversal mapping, 14
singular vector, 283 typical ¬ber, 49
smooth distribution, 24
U
smooth functor, 53
underlying manifold, 169
smooth partition of unity, 5
unitary group, 32
smooth splitting property, 171
universal vector bundle, 97
solvable, 130
source, 117
V
special linear group, 31
vector bundle, 50
special orthogonal group, 31
vector bundle atlas, 50
special unitary group, 32
vector bundle chart, 49
sphere, 9
vector bundle functor on Mfm , 141
spray, 257
vector bundle functor, 56
stable distribution, 26
vector bundle homomorphism, 50
standard ¬ber, 49, 77, 172
vector bundle isomorphism, 50
standard ¬ber of the bundle functor F , 139
vector ¬eld, 16
star bundle functor, 346
vector ¬eld · — dual to a linear vector ¬eld ·,
stereographic atlas, 9
380
Stiefel manifold, 89
vector ¬eld along f , 376
Study numbers, 318
vector sub bundle, 52
submanifold, 9
vector valued di¬erential form, 68
submanifold chart, 9
velocities of order r and dimension m, 120
submersion, 11
vertical bundle, 55, 77, 98
support, 5, 50, 176
vertical lift, 55, 278, 319
support of a vector ¬eld, 19
vertical projection, 55, 78
symplectic group, 31
vertical prolongation, 255
system of standard ¬bers of the bundle
functor, 147 vertical space, 73
vertical Weil bundle, 336
T
W
tangent bundle, 8
tangent space of M at x, 7 W-extendible, 178, 205
tangent vector, 6 weakly local functor, 313
target, 117 weight, 271, 285
tensor evaluation theorem, 224 Weil algebra, 298
tensor ¬eld, 59, 61 Whitney extension theorem, 178
tensorial forms of type , 154 Whitney-extendible, 178
time dependent vector ¬eld, 29 width of a Weil algebra, 299
topological manifold, 4
Z
torsion form, 155
zero section, 50
torsion tensor, 155
torsion-free, 155 zero set, 5




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

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