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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
r
± : J (M, N ) ’ M the source mapping of jets, 12.2
β : J r (M, N ) ’ N the target mapping of jets, 12.2
B : FM ’ Mf the base functor, 2.20
C ∞ E, also C ∞ (E ’ M ) the space of smooth sections of a ¬ber bundle
C ∞ (M, N ) the space of smooth maps of M into N

Cx (M, N ) the space of germs at x ∈ M , 1.4
conja : G ’ G the conjugation in a Lie group G by a ∈ G, 4.24
d usually the exterior derivative, 7.8
the algebra of dual numbers, 37.1
D
Dn = J0 (Rn , R) the algebra of r-jets of functions, 40.5
r r

(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
X
Flt , 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 n-
dimensional ¬bers and locally invertible ¬ber respecting mappings,
17.1

FM the category of star bundles, 41.1
usually a general Lie group with multiplication µ : G — G ’ G, left
G
translation », and right translation ρ
the Lie algebra of a Lie group G
g
r
Gm the jet group (di¬erential group) of order r in dimension m, 12.6
r
the jet group of order r of the category FMm,n , 18.8
Gm,n
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
short for the k — k-identity matrix IdRk
Ik
r
invJ (M, N ) the bundle of invertible r-jets of M into N , 12.3
J rE the bundle of r-jets of local sections of a ¬ber bundle E ’ M , 12.16
r
J (M, N ) the bundle of r-jets of smooth functions from M to N , 12.2
j r f (x), also jx f the r-jet of a mapping f at x, 12.2
r
r
Kn 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


natural numbers
N
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
P M = invJ0 (Rdim M , M ) the r-th order frame bundle of a manifold M , 12.12
r r

πs : J r (M, N ) ’ J s (M, N ) projection of r-jets into s-jets, s ¤ r, 12.2
r

QP the connection bundle of a principal bundle P , 17.4
1
Q„ P M the bundle of torsion free linear connections, 25.3
real numbers
R
r : P — G ’ P usually a right action, in particular the principal right action of
a principal bundle
the tangent bundle of a manifold M with projection πM : T M ’ M
TM
1.7
T M = J r (M, R)— the r-order tangent bundle, 12.14
(r)
0
Tk = J0 (Rk , ) the functor of (k, r)-velocities, 12.8
r r

TA the Weil functor corresponding to the Weil algebra A, 35.11
usually the translation Rm ’ Rm , y ’ y + x
tx
VB the category of vector bundles, 6.3
r
Wm G the (m, r)-principal prolongation of a Lie group G, 15.2
r
WP 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
integers
Z


Author index

Albert, 48 Carrell, 133, 216, 218
Atiyah, 265, 266, 267, 295 Chrastina, 181, 210
Baston, 281, 293, 294, 295 Cohen, 11
Bernstein, 292 Collingwood, 294
Boe, 294 de Le´n, 356
o
Boerner, 131, 281 De Wilde, 210, 252
Boman, 172 Dekr´t, 257
e
Bott, 265, 266, 267, 295 Dieudonn´, 9, 17, 133, 216, 218
e
Branson, 293, 295 Donaldson, 5
Br¨cker, 10
o Doupovec, 229, 359, 375
Cahen, 210 Eastwood, 281, 293, 294
Cap, 252, 254, 393, 416 Eck, 297, 328, 400, 416
430 Author index


Ehresmann, 117, 166, 167, 169, 265 Morrow, 81
Eilenberg, 169 Nagata, 5, 81
Epstein, 116, 168, 188, 210, 295 Naymark, 130, 131, 285
Fegan, 294 Neeb, 48
Feigin, 292 Newlander, 75
Feng Luo, 5 Nijenhuis, 68, 75, 116, 138, 210, 255
Freedman, 4, 5 Nirenberg, 75
Fr¨licher, 59, 75, 79, 396
o Nomizu, 81, 100, 107, 162, 166, 403
Fuks, 292 Ozeki, 81
Gancarzewicz, 345, 357, 416 Palais, 83, 116, 138, 168, 210, 222, 282
Gheorghiev, 167 Patodi, 265, 266, 267, 295
Gilkey, 275, 295 Peetre, 176, 210
Goldschmidt, 340 Pohl, 363
Gompf, 5 Pradines, 334
Graham, 295 Quinn, 4
Greub, 5, 81, 115 Radziszewski, 229, 248
Grozman, 291 Reinhart, 166
Gurevich, 215 Rice, 294
Gutt, 210 Richardson, 68
Halperin, 5, 81, 115 Rodriguez, 356
Hilgert, 48 Rudakov, 286, 288
Hirsch, 10, 11, 82, 180, 314, 330 Sattinger, 48
Hochschild, 201 Saunders, 393
Jacobson, 42 Schouten, 248, 255
Janyˇka, 166, 248, 357, 393
s Sekizava, 277, 295, 357
Joris, 12 Shmelev, 292
J¨nich, 10
a Shtern, 285
Kainz, 297, 328 Stashe¬, 301
Kirillov, 281, 282, 289, 292, 393 Stefan, 48
Kobak, 363 Stefani, 227
Kobayashi, 100, 107, 162, 166, 167, 403 Sternberg, 340
Kock, 349 Stredder, 276, 295
Kowalski, 277, 295 Sussman, 48
Kriegl, 59, 79, 310 Terng, 116, 128, 136, 138, 166, 168, 210, 282,
Krupka, 248, 252, 357, 360 286
Kurek, 232, 265, 402 Thurston, 116, 168, 188, 210,
Laptev, 167 Tougeron, 178
Lecomte, 48, 252, 404 Trautman, 392
Leites, 292 Tulczyjew, 227
Libermann, 115, 167 van Strien, 252
Lichnerowicz, 235, 243 Vanstone, 5, 81, 115
Lubczonok, 248 Varadarajan, 42
Luciano, 297, 328 Virsik, 167
Luna, 224 Vosmansk´, 265, 295, 349
a
Malgrange, 178 Weaver, 48
Mangiarotti, 75, 340 Weil, 296, 301, 328
Mauhart, 48, 60 Weyl, 265
Mikol´ˇov´, 252
as a White, 393
Mikulski, 210, 211, 343, 349, 360 Whitney, 10, 178
Milnor, 4, 301 Wolf, 115
Modugno, 227, 257, 262, 263, 295, 340, 393 Yamabe, 43
Molino, 48 Zajtz, 210, 375
Montgomery, 43, 45, 310 Zhelobenko, 285
Morimoto, 297, 355 Zippin, 43, 45, 310




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Index 431


Index

A cocycle condition, 51, 77
A-admissible, 341 cocycle of transition functions, 51, 77
A-velocity, 306 cohomologous, 51, 87
cohomology classes, 51
absolute di¬erential, 164
complete connection, 81
absolute di¬erential along X, 163, 164
complete vector ¬eld, 19
absolute operator, 351
complete ¬‚ow, 19
action of a category, 147
completely reducible, 131
adjoint representation, 38
conformal, 271
admissible category, 171
conformal weight, 293
a¬ne bundle, 60
a¬ne bundle functor on Mfm , 142 conjugation, 38
connection, 73, 77
algebraic bracket, 68
connection form, Lie algebra valued, 100
algebraic description of Weil functors, 305
connection morphism, 364
almost complex structure, 75
connection, general, 77, 158
almost Whitney-extendible, 184
connector, 110, 326
anholonomic, 16
contact (n, r)-element, 124
associated bundle, 90
cotangent bundle, 61
associated map, 171, 174
covariant derivative, 110, 326
associated map to the k-th order operator A,
143 covariant exterior derivative, 103, 111
associated maps of the bundle functor F , 139 covelocities, 120
atlas, 4 curvature, 73, 111
curvature form, Lie algebra-valued, 100
B
D
base, 11, 50
base extending, 173 derivation, 6, 322
base functor B : FM ’ Mf , 15 derivation, graded, 67
Bianchi identity, 78 derived group, 130
Borel subalgebra, 285 descending central sequence, 130
bundle functor on Mfm , 138 di¬eomorphic, 5
bundle functor on C, 170 di¬eomorphism, 5
bundle functor on the category Mf , 146 di¬erential, 8
di¬erential form, 62
C di¬erential group, 119
C-connection, 365 distinguished chart, 27

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