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Index
A abstract moment map, 141 accessible set, 30, 95, 96 action
adjoint,52 affine, 152 canonical, 136 coadjoint, 52 cotangent lift, 52 effective, 56 faithful, 56 free, 56 globally Hamiltonian, 149 group,51 groupoid,117-119 Hamiltonian, 149 left,51 Lie algebra, 55
canonical, 136 Poisson, 136
Lie group, 51 locally free, 56 Poisson, 136 principal, 65 proper, 59
at a point, 59 at an element, 354
right,51 space, 421 tangent bundle, 54 tangent lift, 52 transitive, 56 tubewise Hamiltonian, 282 twisted,64
action-angle coordinates generalized,421
adjoint
action, 52 group,47 operator, 42 orbit, 56 representation, 39
admissible section, 74 Ado's Theorem, 48 affine
action, 152 connection, 243 Lie-Poisson bracket, 156 Lie-Poisson space, 156
algebra action, 55
canonical, 136 Poisson, 136
antihomomorphism, 55 coboundaries, 49 cocycle,49 cohomology, 49 derivation, 19 Poisson,l24
algebras presheaf of Poisson, 366
algebroid Lie, 116
almost cosymplectic manifold, xxvii amended potential, 238 angular momentum, 145 annihilator, 6, 88, 134 antihomomorphism
Lie algebra, 55 arrow,95
collection, 95 associated bundle, 65, 66 atlas
oriented, 25
478
singular, 33 stratified, 33
automorphism, 47 inner, 42, 47,52 Poisson
infinitesimal, 129 leaf preserving, 134
average of a Riemannian metric, 70
B Baer groupoid, 118 Baire Category Theorem, 9 Baker-Campbell-Hausdorff
formula,40 base
of a groupoid, 116 of a locally trivial fiber
bundle, 16 basis
Hilbert,85 positively oriented, 26
bifoliation, 416 bifurcation lemma, 147, 178 body
coordinates, 128 representation, 219
boundary of a manifold, 26 piece, 31
bracket Lie, 18 Lie-Poisson, 125, 156 Poisson, 124 restricted
Poisson, 125 bundle
associated, 65, 66 chart of a locally trivial fiber
bundle, 16 conormal, 388 cotangent, 4 equations, 293, 298 exterior differential forms, 22 holonomy, 172 horizontal, 171 principal, 65 tangent, 2, 4
C
tensor, 20 vertical, 171
canonical action, 136
Index
coordinates, 122 decomposition associated to
a stratification, 32 distribution, 380 Lie algebra action, 136 map, 126, 127 symplectic form, 122
Cartan formula,24 Theorem, 124
Casimir function, 124 local, 126 reduction, 375
category Hamiltonian covering
map, 182 center, 44 centralizer, 44 characteristic distribution, 126,
211,387 chart, 1
cone,35 foliated, 28, 29 link,35 on the tangent bundle, 3 singular, 32 stratified, 32
compatibility of, 33 Chern class, 173 Chow theorem, 97 Chu momentum map, 141-143 class
Cm
map, 33 Chern,173 homotopy,16
clean intersection, 9 closed
form, 23 locally,31
Index
map,8 co-orientable contact
structure, xxvi coadjoint
action, 52 equivariant
momentum map, 149 isotropy subalgebra, 55 isotropy subgroup, 55 orbit, 56, 216-224 representation, 39, 52
coarse groupoid, 116 coarser decomposition, 31 coboundaries
algebra, 49 coboundary, 50 cocyde,49
algebra, 49 identity, 49, 50 non-equivariance, 151, 153
codimension of a foliation, 28 cohomology, 50
group,50 Lie algebra, 49 Lie group, 49
coisotropic submanifold, 123, 388 subspace, 123 torus, 435
collection of arrows, 95 collective function, 413 commutant
double, 401 commutator
subalgebra, 47 subgroup, 46
compatible function, 13 Jacobi structures, xxv stratified charts, 33
complement symplectic orthogonal, 122
completable, 98 complete
distribution, 98 ftow, 18
foliation symplecticaIly,416
map,414 vector field, 18
completion, 98 complex projective plane, 81 component
isotopic identity, 117
condition Noether, 140 W-spanning, 342, 343 Whitney (A), 34 Whitney (B), 34 Whitney for pairs, 34
cone,34 chart,35 space,35 vertex, 34
conjugation, 42, 52 connected components
group of, 62 connection, 171
affine, 243 ftat, 173 linear, 243 mechanical,238 reducible, 172
connector, 244 conorrnal bundle, 388 conservation of the isotropy, 138 conserved quantity, 125 constant rank map, 5 constraint
Dirac forrnula, 396 first dass, 388 second dass, 392
construction Kaluza-Klein, 235
contact manifold, xxvi one-form, xxvi structure, xxvi
contragredient representation, 52 contravariant tensor, 20 convective representation, 219
479
480
convexity momentum map, 168
coordinates action-angle
generalized,421 body, 128 canonical, 122 Darboux, 122
correspondence theorem
leaf, 409, 430 cosymplectic
manifold, xxvii submanifold, 391
cotangent bundle,4
reduction, 234 lift
action,52 covariant
derivative, 243 tensor, 20
covered space
universal, 182 covering
finite, 16 Hamiltonian, 181 map, 16 space,16
universal, 182 universal, 16 universal group, 48
critical point, 7 value,7
curvature form, 173 curve
horizontal, 171 curves
tangent, 2 cut
symplectic, 169 cylinder valued momentum
map, 176 normal form, 288
D Darboux
coordinates, 122 Theorem, 122
relative, 279 de Rham cohomology group, 24 decomposed
space,31 dimension, 31 morphism, 31 skeleton, 31
subspace, 31 decomposition, 31
associated to a stratification, 32
coarser,31
Index
distribution adapted to a, 380 equivalent, 32 finer, 31 pieces, 31 tangent-normal, 101 Witt-Artin, 271-275
Delzant polytope, 168 depth,31 derivation, 19,47
inner, 47 relative to the wedge
product,23 derivative, 2, 4
covariant, 243 directional, 18 exterior, 23 Lie, 18,20 Lie formula, 19 of a function, 18
determinant Jacobian, 25
diagram reduction
optimal, 440 regular, 231 singular, 329
diffeomorphism, 5, 19 local, 7, 93-94 path,18 Poisson, 127
Index
difference symplectic, 229
differentiable distribution, 29 section
of a distribution, 29 differential structure of a
stratification, 33 dimension, 1
of a decomposed space, 31 of a foliation, 28 of a generalized distribution, 29 of a leaf, 29
Dirac constraints
formula, 396 formula, 398 submanifold, 394
directional derivative, 18 distribution, 27
adapted to a decomposition, 380 canonical, 380 characteristic, 126,211,387 complete, 98 differentiable, 29 generalized, 29 integrable, 27, 30, 380 invariant, 99 involutive, 27, 30 null,211 Poisson, 380 polar, 194 saturating, 108 slice, 386 smooth,380 symplectic orthogona!, 402, 415
divergence, 25 theorem, 27
double commutant relations, 401 Drinfeld, 143
Drinfeld double, 143 dual pair, 401
Howe, 401 Howe localized, 405 Lie-Weinstein,401
singular, 422 von Neumann, 424
duality pairing, 18 Duistermaat-Heckman
function, 320 Theorem, 319
dynamic phase, 255 dynamical system
Poisson, 125 Dynkin formula, 40
E effective
action, 56 potential, 238
Ehresmann Fibration Theorem, 15 eight
figure,7 embedded submanifold, 5 embedding, 5
cotangent bundle reduction, 235
equations bundle, 293, 298 Hamilton, 122 reconstruction, 293, 298
equivalence Morita, 414 of volume forms, 25 relation
graph,lO ofpaths, 16 regular, 10
equivalent decomposition, 32 sets, 31
equivariant function
infinitesimal, 59 infinitesimal
momentum map, 149 mapping,57 momentum map, 149 section, 72, 74 vector field, 99
Euclidean group,60
481
482
locally,43 Eulerian representation, 219 exact
contact manifold, xxvii structure, xxvi
form, 23 exponential
formula, 19 map, 38
extension local, 11,367,381
invariant, 382 property, 58, 86
local, 58, 367, 381 exterior
F
derivative, 23 differential form, 22 product,22
F -topology, 97 faithful action, 56 family
polar, 194 standard, 194
fiber bundle
locally trivial, 15 derivative, 145 of a locally trivial fiber
bundle,16 fibration
cotangent bundle reduction, 239
locally trivial, 15 field
magnetic, 235 figure eight, 7 finer decomposition, 31 finite covering, 16 first
cIass constraint, 388 cohomology group, 50 homotopy group, 16 integral, 28 local integral, 28, 415
fixed point submanifold, 75
infinitesimal, 76 ftat connection, 173 ftow, 17
complete, 18 HamiItonian, 426
foliated chart
generalized, 29 charts,28 manifold, 28
foliation, 27 codimension, 28 dimension, 28 generalized, 29 reduction, 211
Poisson, 387
Index
regular, 28, 212 symplectically complete, 416 tangent bundle, 28
form tangent vector, 28
canonical symplectic, 122 cIosed,23 curvature, 173 exact,23 normal,271
cylinder valued momentum map, 288
Marle-Guillemin-Sternberg, 282,285
volume,24 with values in a vector bundle,
243 formula
Baker-Campbell-Hausdorff, 40 Cartan,24
free
Dirac, 398 Dirac constraints, 396 Dynkin, 40 exponential, 19 Lie derivative, 19 Trotter product, 19
group action, 56
Index
locally action, 56
Freudenthal-Kuranishi-Yamabe theorem, 44
Frobenius theorem, 27, 28, 97
generalized, 30 frontier condition, 31 function
Casimir, 124 collective,413 compatible, 13 Duistermaat-Heckman, 320 equivariant
infinitesimal, 59 germ, 3 Hamiltonian, 122, 124 invariant, 94, 98
infinitesimal, 58 local Casimir, 126 pseudo-Casimir, 375 Whitney,11 Whitney invariant, 57
fundamental group, 16
G g-equivariant function, 59 G-equivariant mapping, 57 G-invariant, 57
G-invariant function, 57
g-invariant function, 58 G-isotopic,95 G-manifold, 51
structure theorem, 83 G-saturated, 57 G-space,51 Gauss Theorem, 27 generalized
action-angle coordinates, 421 distribution, 29 foliated chart, 29 foliation, 29
generator inifinitesimal, 55
geometric phase, 254 germ
as an equivalence dass, 3 of a function, 3 set, 31
global equivariant momentum
map,149 section of a sheaf, 3
globally Hamiltonian action, 149 Godement theorem, 10
483
graph of an equivalence relation, 10 group
action, 51 adjoint,47 cohomology, 49 de Rham cohomology, 24 first homotopy, 16
.fundamental, 16 holonomy, xxviii, 172 Lie,37 loop,48 of connected components, 62 of local Poisson
diffeomorphisms, 127 of Poisson diffeomorphisms, 127 path,48 representation, 52 strongly Hamiltonian, 432 tubewise
strongly Hamiltonian, 432 weakly Hamiltonian, 432
universal covering, 48 von Neumann, 424 weakly Hamiltonian, 432
groupoid, 115-119 action, 117-119 Baer,118 base, 116 coarse,116 homomorphism, 117 integrable, 117 inversion map, 116 isotropy subgroup, 117 Lie,116 orbit, 117 pair, 116 product, 116
484
H
symplectic, 118 total space, 116 transformation, 117 transitive, 117 trivial, 116
Haar measure, 41 Hamilton equations, 122 Hamiltonian
action, 149 tubewise, 282
covering, 181 covering category, 182 covering map, 181 ftow, 426 function, 122, 124 group
strongly, 432 tubewise strongly, 432 tubewise weakly, 432 weakly, 432
locally, 123, 130 quasi,188 reduced,207,225,303,323,364 stages hypothesis, 361 system, 122
noncommutative integrable, xxxii,421
reduced, 207 super integrable, 421 superintegrable, xxxii symmetric, 136
universal covered space, 182 covering space, 182
vector field, 122, 124, 130 Hilbert
basis, 85 Fifth Problem, 43 map, 85 theorem, 85
holonomy bundle,l72 group, xxviii, 172 theorem, 173
homogeneous
manifold, 56, 62 space, 56, 62
homomorphism ofLie algebras, 41 ofLie groups, 41 of groupoids, 117
homotopy, 16 c1asses, 16
horizontal bundle, 171 curve, 171 lift, 171 lift operator of a linear
connection, 244 subbundle of a linear
connection, 244 vector, 171
Howe dual pair, 401 pair
localized,405 singular, 422
hyper-Kähler manifold, xxviii hypothesis
I
stages, 257, 355 Hamiltonian, 361
ideal,45 identity
cocyc1e,49 section, 116 two-cocyc1e, 50, 153
immersed submanifold, 5 immersion, 5
regular, 5 injective, 5 stratified, 32
incident, 31
Index
induced boundary orientation, 26 inertia
tensor locked,238
infinitesimal equivariant
function, 59 momentum map, 149
Index
fixed point manifold, 76 generator, 55 invariant
funetion, 58 isotropy type, 76 orbit type, 76 Poisson automorphism, 129
preserving leaves, 134 type manifolds, 76
initial smooth strueture of a saturated
set, 109 submanifold, 5 topology, 320
injeetive immersion, 5 inner
automorphism, 42, 47, 52 derivation, 47
integrable distribution, 27, 30, 380 groupoid, 117 noneommutative, 420, 421 pseudogroup, 95
integral first, 28 loeal
first, 28, 415 loeal manifold, 27 manifold, 29
maximal,30 ofa form, 25 of a funetion, 25 of the motion, 125
interior of a manifold, 26 produet,24
invariant distribution, 99 funetion, 57, 94, 98
infinitesimal, 58 funetions
presheaf, 11 loeal
extension, 382 set, 11,57,94,98 veetor,76
Whitney funetion, 57
Whitney funetion, 12 inversion map of a groupoid, 116 involutive, 27
distribution, 30 funetions, 390
isomorphism sympleetie, 122
isotopic, 95 eomponent
identity, 117 groupoid elements, 117
isotopy lemma, 35 isotropie
submanifold, 123 subspaee, 123
isotropy
J
eonservation law, 138 subalgebra, 55, 162 subgroup, 55
groupoid, 117 type, 75
infinitesimal,76 loeal,82
Jaeobi eompatible
structures, xxv identity, 18,47 manifold, xxiii
transitive, xxiv
485
J aeobi-Lie-Caratheodory Theorem, 124
Jaeobi-Nijenhuis manifold, xxv Jaeobian determinant, 25
K Kaluza-Klein
eonstruetion, 235 system, 236
kernel of a Lie group homomorphism, 43
KKS sympleetie form, 132, 158 Kostant-Kirillov-Souriau sympleetie
form, 132, 158
486
Kähler locally conforrnal
manifold, xxviii manifold, xxviii, 301
Kähler-Einstein manifold, xxviii
L Lagrangian
function, 145 nondegenerate, 145 regular, 145 representation,219 submanifold, 123 subspace, 123
law of conservation of the isotropy, 138
leaf
left
correspondence theorem, 409, 430
of a foliation, 27 preserving Poisson
automorphism, 134 regular, 29 singular, 29 space of, 28 symplectic, 130,427 topology
symplectic, 428
action, 51 invariant vector field, 38 translation, 52 trivialization, 54, 128
Legendre transforrn, 145 Leibniz
identity, 19 manifold, 363
lemma
Lie
bifurcation, 147, 178 Poincare, 23 reduction, 161, 162 Relative Poincare, 23 Second Whitehead, 151 Whitehead, 49, 50
algebra, 19
action, 55 antihomomorphism, 55 canonical action, 136 cohomology, 49 homomorphism, 41 of a Lie group, 38 Poisson action, 136
algebroid, 116 bracket, 18 derivative
forrnula, 19 on functions, 18 on tensor fields, 20 on vector fields, 18
group,37 action, 51 canonical action, 136 cohomology, 49 homomorphism, 41 Poisson action, 136 tangent bundle, 54
Index
valued momentum map, 188 groupoid, 116 subgroup, 43 Third Fundamental Theorem, 48
Lie-Poisson bracket, 125, 156
affine, 156 space
affine, 156 structure, 156
Lie-Weinstein dual pair, 401 lift
cotangent action,52
horizontal, 171 tangent
action, 52 Lifted group actions, 60 Lindelöf topological space, 9 linear
connection, 243 horizontal lift operator, 244 horizontal subbundle, 244
momentum, 145 representation, 52
Index
symplectic map, 122
link,35 chart,35
Liouville vector field, 250 Liouville-Mineur-Arnold Theorem,
420 local
Casimir function, 126 diffeomorphism, 7, 93-94
Poisson, 127 extension, 11,367,381
property, 58, 367, 381 first integral, 28, 415 integral manifold, 27 invariant
extension, 382 isotropy type, 82 operator, 18 orbit type, 82 representative of a map, 2 slice, 28
of a regular distribution, 386 type manifold, 82
localized Howe dual pair, 405 locally
closed,31 subset, 2
compact topological space, 2 conformal
Kähler manifold, xxviii symplectic manifold, xxvi
EUclidean, 43 trivial fiber bundle, 15
bundle chart, 16 fiber, 16
trivial fibration, 15 locked inertia tensor, 238 logarithm, 38 loop group, 48
M magnetic
field,235 potential, 236 term, 235
manifold
map
3-Sasakian, xxix almost cosymplectic, xxvii boundary, 26 contact, xxvi cosymplectic, xxvii exact contact, xxvii fixed point, 75
infinitesimal, 76 foliated, 28 homogeneous, 56,62 hyper-Kähler, xxviii integral, 29 interior, 26 isotropy type, 75
infinitesimal, 76 Jacobi, xxiii
transitive, xxiv Jacobi-Nijenhuis, xxv Kähler, xxviii, 301 Kähler-Einstein, xxviii Leibniz, 363 locally conformal
Kähler, xxviii symplectic, xxvi
orbit type, 75 infinitesimal, 76
orientable, 24 oriented, 24 Poisson, 124 Poisson-Nijenhuis, xxiv presymplectic, 122,211 quaternion-Kähler, xxviii Sasakian, xxix simply connected, 16 symplectic, 121 toric, 168 type, 75
infinitesimal, 76 local,82
V,319 volume,24
canonical, 126, 127 closed,8 complete, 414 constant rank, 5
487
488
covering, 16, 17 Hamiltonian covering, 181 Hilbert,85 inversion of a groupoid, 116 moment, 118, 121, 144 momentum, 121, 196
Chu, 141-143 cylinder valued, 176 Lie group valued, 188 Noether, 140 optimal, 196 standard, 144
of class Cm , 33 open, 7 orientation
preserving, 25 reversing, 25
Poisson, 126 product, 116 proper, 8, 59 smooth, 2, 4 stratified, 32 symplectic, 127
linear, 122 tangent, 2, 4 transversal, 8 volume preserving, 25
Marle-Guillemin-Sternberg normal form, 282, 285
Marsden-Weinstein reduced space, 206
material representation, 219 maximal integral manifold, 30 measure
Haar, 41 mechanical
connection, 238 system
simple, 238 simple with symmetry, 238
metric Riemannian, 20
model space, 1 moment
map, 118, 121, 144 abstract, 141
momentum angular, 145 linear, 145 map, 121, 196
Chu, 141-143 convexity, 168
Index
cylinder valued, 176 equivariant, 149 infinitesimally equivariant,
149 Lie group valued, 188 Noether, 140 optimal, 196 reduced, 176 standard, 144
polytope, 168 space, 196 split,299
monoid,94 Morita equivalence, 414 morphism
N
of decomposed spaces, 31 of stratified spaces, 32
Neil parabola, 35 Nijenhuis
operator, xxiv, xxv torsion, xxiv, xxv
Noether condition, 140 momentum map, 140 theorem, 146
non-equivariance cocycle, 151, 153 noncommutative
integrability, 420, 421 integrable Hamiltonian system,
xxxii, 421 nondegenerate
Lagrangian, 145 presheaf of Poisson algebras, 366
normal bundle to a foliation, 28 form, 271
cylinder valued momentum map,288
Index
Marle-Guillemin-Sternberg, 282,285
subgroup, 45 sympleetie spaee, 276
normalizer, 44, 45 null distribution, 211
o one
form, 20 eontaet, xxvi
parameter subgroup, 42 one-eoboundary, 50 one-eoeycle, 49 open map, 7 operator
adjoint,42 loeal, 18 Nijenhuis, xxiv, xxv reeursion, xxiv
optimal momentum map, 196 orbit
redueed spaee, 338 reduetion, 337
point redueed spaee, 332 reduetion
diagram, 440 Sjamaar principle, 337
orbifold, 319 orbit
adjoint,56 eoadjoint, 56, 216--224 groupoid, 117 of a group action, 55 of a pseudogroup, 94 prineipal, 84 redueed spaee, 224
optimal, 338 reduetion, 224-231, 322-323
optimal,337 stratifieation theorem, 329 sympleetie, 373
regular, 84 regularization, 339 spaee, 56, 94 sympleetie form, 132, 158
type, 75 infinitesimal, 76 loeal,82
orientable manifold, 24 orientation, 25
indueed boundary, 26 preserving map, 25 reversing map, 25
oriented atlas, 25 manifold, 24
orthogonal
p pair
sympleetie, 122 distribution, 402, 415
dual,401 Howe,401 Lie-Weinstein, 401 singular, 422 von Neumann, 424
groupoid, 116 Howe
loealized, 405 singular, 422
pairing, 18 duality, 18
parabola Neil,35
paraeompaet topologieal spaee, 13 partition of unity, 13 path
equivalenee of, 16 group,48 of diffeomorphisms, 18
Pfaffian system, 99 phase
dynamie, 255 geometrie, 254 reeonstruetion, 254
piece boundary, 31 of a deeomposition, 31
plane projeetive, 81
Poineare lemma, 23
489
490
relative, 23 point
critical,7 principal, 84 reduced space, 206, 302
optimal, 332 reduction
optimal, 331 regular, 7, 29 singular, 7, 29
point reduction, 206-216, 302-303 Poisson
action, 136 algebra, 124
nondegenerate, 366 algebras
presheaf, 366 automorphism
infinitesimal, 129 leaf preserving, 134
bracket, 124 restricted, 125
diffeomorphism, 127 local, 127
distribution, 380 dynamical system, 125 foliation reduction, 387 Lie algebra action, 136 manifold, 124 map, 126 reducible, 368, 382 reduction, 363
regular, 364 structure
transverse, 126, 399 submanifold, 128
quasi-, 128 tensor, 125
Poisson-Nijenhuis manifold, xxiv polar
distribution, 194 family, 194
standard, 194 pseudogroup, 194
standard, 194 reduced space, 340, 341
regularized, 342 reduction, 331
polarity, 194 polytope
Delzant, 168 momentum, 168 rational, 168 simple, 168 smooth,168
positively oriented basis, 26 potential
amended, 238 effective, 238 magnetic, 236
presheaf,3 of invariant functions, 11 of Poisson algebras, 366
nondegenerate, 366 of saturated functions, 11
Index
of smooth G-invariant functions, 57
of smooth functions, 4, 33 of Whitney invariant functions,
12,57 product,13 quotient, 11
presymplectic manifold, 122,211 principal
action, 65 bundle,65 fiber bundle reduction, 172 map, 66 orbit, 84 orbit theorem, 84 point, 84
principle Sjamaar, 309, 336
optimal, 337 product
exterior, 22 groupoid, 116 interior, 24 map, 116 of groups
semidirect, 53 of Lie algebras
Index
semidirect, 54 presheaf, 13 semidirect, 53-54 twisted,64 wedge,22
projective plane, 81 proper
action, 59 at a point, 59 at an element, 354
map, 8,59 property
submanifold, 2 extension, 58, 86
local,58 pseudo-Casimir function, 375 pseudogroup, 94-95
integrable, 95 orbit, 94 polar, 194
standard, 194 von Neumann, 424
pull-back, 19 push-forward, 19
offorms, 23
Q quadrature, 215 quasi-
Hamiltonian space, 188 Poisson submanifold, 128
quatemion-Kähler manifold, xxviii quotient
presheaf, 11 topology, 10
R range of a Lie group homomorphim,
43 rank, 126
of a generalized distribution, 29 of a point, 33
rational polytope, 168 reconstruction
equations,293,298 of a solution, 214 phase, 254
recursion operator, xxiv reduced
Hamiltonian, 207, 225, 303, 323,364
system, 207 momentum map, l76 space
Marsden-Weinstein, 206 optimal, 332 orbit, 224, 322 point, 206, 302 polar, 340, 341 regularized, 342 symplectic, 206
system, 133 reducible
connection, 172 Poisson, 368, 382
reduction, 205 by stages, 266 Casimir, 375 diagram
optimal, 440 regular, 231 singular, 329
foliation, 211 lemma, 161, 162
491
of a principal fiber bundle, 172 of cotangent bundle
at zero, 234 embedding, 235 fibration, 239
optimal orbit, 337 point, 331
orbit, 224-231, 322-323 point, 206-216, 302-303 Poisson, 363
foliation, 387 regular, 364
polar, 331 symplectic, 205, 211-213, 373
orbit, 224-231 point, 206-216
theorem, 172 universal, 374
492
regular (A)-pair, 34 (B)-pair, 34 distribution
slice, 386 equivalence relation, 10 foliation, 28, 212 immersion, 5 Lagrangian, 145 leaf,29 Lie subgroup, 43 orbit, 84 point, 7, 29 Poisson reduction, 364 reduction
diagram, 231 value,7
regularization, 310, 326, 337 orbit, 339
regularized polar reduced space, 342
related vector fields, 21 relation
double commutant, 401 Relative
Darboux Theorem, 279 Poincare Lemma, 23
representation, 49, 52 adjoint, 39, 52 body,219 coadjoint, 39, 52 contragredient, 52 convective,219 Eulerian,219 group,52 Lagrangian, 219 linear, 52 material, 219 space,219
residual, 9 restricted Poisson bracket, 125 restriction
morphism,3 of a subbundle, 27
retraction, 67 Riemannian metric, 20
Index
invariant, 70 Riesz Representation Theorem, 26 right
S
action, 51 invariant vector field, 38 translation, 52 trivialization, 54
Sard Theorem, 9 Sasakian manifold, xxix saturated
functions presheaf, 11
set, 11,57,94,98 initial smooth structure, 109
saturating distribution, 108 saturation of a set, 57 second
class constraint, 392 countable topological space, 9 Whitehead Lemma, 151
section admissible,74 equivariant, 72, 74 global
of a sheaf, 3 identity, 116 of a sheaf, 3 of a vector bundle, 243
semidirect product of groups, 53 product of Lie algebras, 54 products, 53-54
separations of orbits, 57 set
sets
accessible, 95, 96 germ, 31 invariant, 11, 57, 94, 98 saturated, 11, 57, 94, 98 saturation, 57
equivalent, 31 sheaf,3
global section, 3 section,3 smooth map, 4
Index
stalk,3 shifting theorem, 233 simple
mechanical system, 238 mechanical system with
symmetry, 238 polytope, 168
simply connected, 16 singlet,76 singular
atlas, 33 chart,32 dual pair, 422 Howe pair, 422 leaf,29 point, 7, 29 reduction diagram, 329
Sjamaar principle, 309, 336
optimal, 337 skeleton of a decomposed space, 31 slice, 67-71
for an equivalence relation, 10 local,28
of a regular distribution, 386 theorem, 71
symplectic, 271 smooth,33
distribution, 380 functions
presheaf, 4, 33 map, 2 map for sheaves, 4 polytope, 168 stratified space, 33 structure of a stratification, 33
source, 116 space
action, 421 cone,35 covering, 16 decomposed, 31 homogeneous, 56,62 locally trivial stratified, 34 model,1 momentum, 196
of invariant vectors, 76 of leaves, 28 of one-forms, 20 of orbits, 56, 94 of singlets, 76 orbit reduced, 224
optimal, 338 point reduced, 206 polar
reduced, 340, 341 quasi-Hamiltonian, 188 reduced
optimal, 332 regularized, 342
representation, 219 smooth stratified, 33 Sternberg, 242 stratified, 32 symplectic normal, 276 tangent, 2, 4 total, 116 universal
covered, 182 covering, 182
universal covering, 16 Weinstein, 242 Whitney stratified, 34
split momentum value, 299 stability subgroup, 55 stabilizer, 55 stages
hypothesis, 257, 355 Hamiltonian, 361
reduction, 266 stalk of a sheaf, 3 standard
momentum map, 144 polar family, 194 polar pseudogroup, 194
Stefan-Sussmann Theorem, 30 Sternberg space, 242 Stokes Theorem, 27 stratification, 32
493
canonical associated decomposition, 32
theorem, 84-85, 301, 315-318
494
orbit reduetion, 329 stratified
atlas, 33 ehart,32 immersion, 32 map, 32 spaee,32
loeally trivial, 34 smooth,33
subimmersion, 32 submersion, 32
stratum, 32 sympleetie
orbit, 322-323 point, 302-303
strong Hamiltonian group, 432 strueture
Lie-Poisson, 156 theorem, 309, 326 theorem for Abelian Lie groups,
39 theorem of G-manifolds, 83 transverse Poisson, 126, 399
subalgebra eommutator,47 isotropy, 55, 162
subbundle vertieal, 244
subgerm,31 subgroup
eommutator, 46 isotropy, 55
groupoid, 117 Lie, 43 normal,45 one parameter, 42 regular, 43 stability, 55
subgroupoid, 117 wide, 117
subimmersion, 5 at a point, 8 stratified, 32
submanifold, 2 eoisotropie, 123, 388 eosympleetie, 391
Dirae, 394 embedded,5 fixed point, 75
infinitesimal, 76 immersed,5 initial,5 isotropie, 123 isotropy type, 75
infinitesimal, 76 Lagrangian, 123 orbit type, 75
infinitesimal, 76 Poisson, 128
quasi-, 128 property,2 sympleetie, 123 type, 75
infinitesimal, 76 submanifolds
transversal,9 submersion, 5
stratified, 32 subordinate to an atlas, 13 subspace
sum
eoisotropie, 123 deeomposed, 31 isotropie, 123 Lagrangian, 123 sympleetie, 123
sympleetie, 229
Index
superintegrable Hamiltonian system, xxxii,421
symmetrie Hamiltonian system, 136 simple meehanieal system, 238
sympleetie cut, 169 differenee, 229 Foliation Theorem, 130, 334 form
eanonieal, 122 orbit, 132, 158
groupoid, 118 isomorphism, 122 KKS form, 132, 158
Index
leaf, 130, 427 correspondence theorem, 409,
430 topology, 428
linear map, 122 locally conformal manifold, xxvi manifold, 121 map,127 normal space, 276 orbit reduction, 224-231,
322-323 orbit stratum, 322-323 orthogonal
complement, 122 distribution, 402, 415
point reduction, 206-216, 302-303
point stratum, 302-303 reduced space, 206 reduction, 205, 211-213, 373
orbit, 373 slice theorem, 271 stratification theorem, 301,
315-318 submanifold, 123 subspace, 123 sum,229 tube, 276
symplectically complete foliation, 416 symplectomorphism, 127 system
T
Hamiltonian, 122 Kaluza-Klein, 236 Pfaffian, 99 reduced, 133
tangent bundle, 2, 4
action, 54 chart,3 Lie group, 54 of a foliation, 28
curves,2 lift
action, 52 map, 2, 4
space, 2, 4 vector,2
field to a submanifold, 20 to a foliation, 28
495
target, 116 tensor
term
bundle,20 contravariant, 20 covariant, 20 inertia
locked,238 Poisson, 125
magnetic, 235 theorem
Ado,48 Baire category, 9 Cartan, 124 Chow,97 Darboux, 122
relative, 279 Divergence, 27 Duistermaat-Heckman, 319 Ehresmann Fibration, 15 Freudenthal-Kuranishi-Yamabe,
44 Frobenius, 27, 28, 97 Gauss,27 Godement, 10 Hilbert,85 holonomy, 173 Jacobi-Lie-Caratheodory, 124 Lie's Third, 48 Liouville-Mineur-Arnold,420 Noether, 146 principal orbit, 84 reduction, 172 Riesz Representation, 26 Sard,9 shifting, 233 slice, 71 Stefan-Sussmann, 30 Stokes, 27 stratification, 84-85
orbit reduction, 329 structure, 309, 326
496
of G-manifolds, 83 structure for Abelian Lie groups,
39 symplectic foliation, 130, 334 symplectic leaf correspondence,
409,430 symplectic slice, 271 symplectic stratification, 301,
315-318 Transport, 26 transversal mapping, 9 tube,69
Thom isotopy lemma, 35 three-Sasakian manifold, xxix time-dependent
fiow,18 vector field, 18
topological space Lindelöf,9 locally compact, 2 paracompact, 13 quotient, 10 second countable, 9
topology F,97 initial, 320 quotient, 10 symplectic leaf, 428
toric manifold, 168 torsion
Nijenhuis, xxiv, xxv torus
coisotropic, 435 transformation groupoid, 117 transitive
group action, 56 groupoid, 117 Jacobi manifold, xxiv
translation left,52 right,52
Transport Theorem, 26 transversal
mapping theorem, 9 submanifolds, 9 to a submanifold, 8
Index
transverse Poisson structure, 126,399 trivial
groupoid, 116 stratified space, 34
trivialization left, 54, 128 right,54
Trotter product formula, 19 tube,67
symplectic, 276 theorem, 69
tubewise Hamiltonian action, 282 strong Hamiltonian group, 432 weak Hamiltonian group, 432
twisted action, 64 product,64
two-cocycle identity, 50, 153 type
u
manifold, 75 infinitesimal, 76 local,82
submanifold, 75 infinitesimal, 76
universal covering, 16
group,48 Hamiltonian
V
covered space, 182 covering space, 182
reduction, 374
V-manifold,319 value
critical,7 regular, 7
vector bundle
section, 243 valued forms, 243
field, 17 complete, 18 divergence, 25
Index
equivariant, 99 Hamiltonian, 122, 124, 130 left invariant, 38 Liouville, 250 locally Hamiltonian, 123, 130 right invariant, 38 tangent to a submanifold, 20 time-dependent, 18
fields related,21
horizontal, 171 invariant, 76 tangent, 2 vertical, 171
vertex of a cone, 34 vertical
bundle,171 lift, 244 subbundle, 244 vector, 171
volume form, 24
equivalent, 25 manifold, 24 preserving map, 25
von Neumann
W
dual pair, 424 group,424 pseudogroup,424 weakly,343
W-spanning condition, 342, 343 weak Hamiltonian group, 432 weakly von Neumann group, 343 wedge product, 22 Weinstein space, 242 Whitehead Lemma, 49, 50 Whitney
condition (A), 34 condition (B), 34 condition for pairs, 34 functions, 11 invariant functions, 12, 57 space,34
wide subgroupoid, 117 Witt-Artin decomposition, 271-275
497
Progress in Mathematics Edited by:
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150 SHIOTA. Geometry of Subanalytic and 161 SAGAN/STANLEY (eds). Mathematieal Semialgebraic Sets Essays in Honor of Gian-Carlo Rota
151 HUMMEL. Gromov's Compactness Theorem 162 ARNoLD/GREuErJSTEENBRINK. Singularities: for Pseudo-holomorphie Curves The Brieskorn Anniversary Volume
152 GROMOV. Metrie Struetures for Riemannian 163 BERNDT/SCHMIDT. Elements of the and Non-Riemannian Spaces. Representaiton Theory ofthe Jacobi Group Sean Bates (translator) 164 ROUSSARIE. Bifurcations ofPlanar Vector
153 BUESCU. Exotic Attractors: From Fields and Hilbert's Sixteenth Problem Liapunov Stability to Riddled Basins 165 MIGLIORE. Introduction to Liaison Theory
154 BÖTICHERIKARLOVICH. Carleson Curves, and Deficiency Modules Muckenhoupt Weights, and Toeplitz 166 ELIAS/GIRAu'MJRO-ROIGIZARZUELA (eds). Operators Six Leetures on Commutative Algebra
155 DRAGOMIRIORNEA. Loeally Conformal 167 FAccHINI. Module Theory Kähler Geometry 168 BALOalKATONAlSZA'szlREcSKI (eds).
156 GUIV ARC'HlJlfI' A YLOR. Compaetifieations European Congress of Mathematies, of Symmetrie Spaees Budapest, July 22-26, 1996. Vol. I
157 MURTY!MURTY. Non-vanishing of 169 BALOG/KATONAlSZA'SZlRECSKI (eds). L-funetions and Applieations European Congress of Mathematies,
158 TIRAONoGANlWoLF (eds). Geometryand Budapest, July 22-26, 1996. Vol. 11 Representation Theory of Real and 168/169 Sets Vols I, 11 p-adie Groups 170 PATERSON. Groupoids, Inverse
159 THANGA VELU. Harmonie Analysis on the Semigroups, and their Operator Algebras Heisenberg Group 171 REZNIKov/SCHAPPACHER (eds).
160 KASHIwARAIMATsuo/SAITO/SATAKE (eds). Regulators in Analysis, Geometry Topologieal Field Theory, Primitive Porms and Number Theory and Related Topies 172 BRYLINSKIlBRYLINSKI!NISTORfI'SYGANI
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174 GoERss/JARDINE. Simplicial Homotopy Theory
175 BANUELOSIMOORE. Probabilistic Behavior of Harmonie Functions
176 BAsslLuBOTZKY. Tree Lattices 177 BIRKENHAKEILANGE. Complex Tori 178 PUIG. On the Local Structure of Morita
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179 MORALES RUIZ. Differential Galois Theory and Non-Integrability of Hamiltonian Systems
180 PATERNAIN. Geodesie Flows 181 HAUSERILIPMAN/OORT/QUIRÖS (eds).
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182 BILLEYILAKSHMIBAJ. Singular Loci of Schubert Varieties
183 KApOVICH. Hyperbolic Manifolds and Discrete Groups
184 du SAUTOy/SEGALlSHALEV (eds). New Horizons in pro-p Groups
185 FARAUT/KANEYUKI/KoRANYIlLulRoos. Analysis and Geometry on Complex Homogeneous Domains
186 KLEINERT. Units in Skew Fields 187 BOSTILoESERlRAYNAUD. Courbes
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188 DOLBEAULT!IORDANIHENKIN/SKODAI TREPREAU (eds). Complex Analysis and Geometry: International Conference in Honor of Pierre Lelong
189 ANDREIBALDASSARJ. DeRham Cohomology of Differential Modules on Aigebraie Varieties
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196 AGUADElBROToICASACUBERTA (eds). Cohomological Methods in Homotopy Theory
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206 SNAITH. Aigebraic K-groups as Galois Modules
207 LONG. Index Theory for Symplectic Paths with Applications
208 TURAEV. Torsions of 3-dimensional Manifolds
209 UNTERBERGER. Automorphic Pseudodifferential Analysis and Higher Level Weyl Calculi
210 JOSEPHIMELNIKOV/RENTSCHLER (eds). Studies in Memory of Issai Schur
211 HERNANDEZ-LERMAILASSERE. Markov Chains and Invariant Probabilities
212 LUBOTZKY/SEGAL. Subgroup Growth 213 DUVALlGuIEu/OvSIENKO (cds). The Orbit
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214 DUNGEY/TER ELSTIROBINSON. Analysis on Lie Groups with Polynomial Growth
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217 THANGAVELU. An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie Groups.
218 FRESENEUV AN DER PUT. Rigid Analytical Geometry and its Applications.
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