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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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