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orem, 170
system, 109, 110, 172
Hamilton equations, 23, 24, 107, 113, integral
123, 148 curve, 106, 113, 127
Hamilton-Jacobi equations, 105 ¬rst, 109
hamiltonian of motion, 109, 147
action, 129, 130, 133, 164 intersection of lagrangian submani-
function, 105, 106, 109, 134 folds, 55
G-space, 134 inverse square law, 113
mechanics, 105 isometry, 120
moment map, 134 isotopy
reduced, 148 de¬nition, 35
system, 109 symplectic, 42
212 INDEX


vs. vector ¬eld, 35 generating function, 17, 23
isotropic intersection problem, 55
of T — X, 16
embedding, 53
vs. symplectomorphism, 15, 19
subspace, 8
zero section, 16
isotropy, 135
lagrangian subspace, 8, 46, 77
J-anti-holomorphic function, 80 Laplace-Beltrami operator, 98
(J-)anti-holomorphic tangent vectors, laplacian, 98
78 Lebesgue
J-holomorphic curve, 82 measure, 191
J-holomorphic function, 80, 82 volume, 191
(J-)holomorphic tangent vectors, 78 left multiplication, 130
Jacobi left-invariant, 130
Hamilton-Jacobi equations, 105 Legendre
identity, 108, 139 condition, 117
jacobiator, 139 transform, 121, 122, 125, 126
Leibniz rule, 109, 139
K¨hler
a Lie
compact K¨hler manifolds, 98
a algebra, 108, 130, 164
form, 90, 98 algebra cohomology, 165
local form, 94 bracket, 108
manifold, 90, 98 derivative, 36, 40
potential, 93, 94 group, 128
recipe, 92 Lie-Poisson symplectic form, 139, 150
submanifold, 94 lift
Killing form, 158 of a di¬eomorphism, 11
kinetic energy, 112, 113 of a path, 115, 119
Kirillov of a vector ¬eld, 106
Kostant-Kirillov symplectic form, linear momentum, 137
139, 150 Liouville
Kodaira Arnold-Liouville theorem, 110
complex surface, 103 form, 13
complex surfaces, 102 measure, 191
Kodaira-Thurston example, 102 torus, 110
Kostant-Kirillov symplectic form, 139, local form, 35, 94, 192
150 locally free action, 135

Lagrange manifold
Euler-Lagrange equations, 120 almost symplectic, 74
variational principle, 114 complex, 83
lagrangian complement, 47 in¬nite-dimensional, 158
lagrangian ¬bration, 111 K¨hler, 90, 98
a
lagrangian submanifold of contact elements, 61
closed 1-form, 17 of oriented contact elements, 62
conormal bundle, 18 riemannian, 119
de¬nition, 16 symplectic, 6
213
INDEX


toric, see toric manifold theorem “ local version, 45
with corners, 188 theorem “ version I, 43
Marsden-Weinstein-Meyer theorem “ version II, 44
quotient, 141 trick, 42“44, 50
theorem, 136, 141 motion
Maupertius constant of motion, 109
variational principle, 114 equations, 113
McDu¬ counterexample, 43 integral of motion, 109, 147
measure
neighborhood
Duistermaat-Heckman, 191
convex, 37
Lebesgue, 191
µ-neighborhood theorem, 38
Liouville, 191
tubular neighborhood, 51
symplectic, 191
tubular neighborhood ¬bration,
mechanical system, 113
39
mechanics
tubular neighborhood in Rn , 41
celestial, 33
tubular neighborhood theorem,
classical, 107
37
metric, 24, 70, 119
Weinstein lagrangian neighbor-
Meyer, see Marsden-Weinstein-Meyer
hood, 46, 48
minimizing
Weinstein tubular neighborhood,
action, 115
51
locally, 115, 117
Newlander-Nirenberg theorem, 82,
property, 117
88
moduli space, 159
Newton
moment map
polytope, 177
actions, 127
second law, 105, 107, 113, 114
de¬nition, 133
Nijenhuis tensor, 82, 88
e¬ective, 176
Nikodym
equivariance, 134
Radon-Nikodym derivative, 191
example, 162
Nirenberg
existence, 164, 166
Newlander-Nirenberg theorem,
hamiltonian G-space, 134
82
in gauge theory, 155
Noether
origin, 127
principle, 127, 147
uniqueness, 164, 167
theorem, 147
upgraded hamiltonian function,
non-singular projective variety, 95
130
nondegenerate
moment polytope, 170
bilinear map, 4
momentum, 107, 123, 137
¬xed point, 55
momentum vector, 137
normal
Morse
bundle, 37, 41
Morse-Bott function, 175
space, 37, 41, 51
Morse function, 55
number
Morse theory, 55, 170
Betti, 100
Moser
equation, 44 Hodge, 100
214 INDEX


one-parameter group of di¬eomor- simple, 177
phisms, 127, 128 smooth, 177
operator positive
Laplace-Beltrami, 98 form, 92
orbifold inner product, 24, 77
conehead, 150 vector ¬eld, 66
dunce cap, 150 potential
examples, 150 energy, 112, 113
reduced space, 150 gravitational, 113
teardrop, 150 K¨hler, 93, 94
a
orbit strictly plurisubharmonic, 92
de¬nition, 135 primitive vector, 178
point-orbit projection, 135 principal bundle
space, 135 connection, 155
topology of the orbit space, 135 gauge group, 158
unstable, 135 principle
oriented surfaces, 50 Noether, 127, 147
overtwisted contact structure, 66 of least action, 114
variational, 114
pendulum product group, 149
simple, 112, 114 projectivization, 61
spherical, 152 proper function, 15, 103, 121
periodic point, 29 pseudo-holomorphic curve, 67, 82
phase space, 107, 113, 148 pullback, 7
Picard theorem, 36
Poincar´e quadratic growth at in¬nity, 126
last geometric theorem, 33 quadrature, 153
Poincar´-Birkho¬ theorem, 33
e quotient
recurrence theorem, 32 Hausdor¬, 136
point-orbit projection, 135 Marsden-Weinstein-Meyer, 141
Poisson symplectic, 141
algebra, 109 topology, 135
bracket, 108, 109, 134, 164
Radon-Nikodym derivative, 191
Lie-Poisson symplectic form, 139,
rank, 4
150
structure on g— , 139 rational polytope, 177
recipe
polar decomposition, 69, 71
for K¨hler forms, 92
a
polytope
for symplectomorphisms, 22
Delzant, 177, 189
recurrence, 29, 32
example of Delzant polytope, 177
reduced
example of non-Delzant poly-
hamiltonian, 148
tope, 178
facet, 178 phase space, 148
moment, 170 space, 136, 141, 150
Newton, 177 reduction
rational, 177 example, 169
215
INDEX


for product groups, 149 moduli, 159
in stages, 149 normal, 41, 51
of connections, 158
local form, 192
phase, 107, 113
low-brow proof, 141
total, 155
Noether principle, 147
spherical pendulum, 152
other levels, 149
splittings, 78
preview, 136
stability
reduced space, 136
de¬nition, 121
symmetry, 147
set, 122
Reeb vector ¬eld, 63
stabilizer, 135
representation
stable
adjoint, 130, 131
function, 125
coadjoint, 130, 131
point, 112, 152
of a Lie group, 128
Stein manifold, 103
retraction, 40
stereographic projection, 89, 97
Riemann
Sternberg
Cauchy-Riemann equations, 84
Atiyah-Guillemin-Sternberg the-
surface, 103, 158
orem, 170
riemannian
Stokes theorem, 13, 161
distance, 25
strictly convex function, 117, 121,
manifold, 24, 119
125
metric, 24, 49, 70, 119
strictly plurisubharmonic, 92
right multiplication, 130
strong isotopy, 42, 50
right-invariant, 130
submanifold, 15
s.p.s.h., 92 submanifold
Seiberg-Witten invariants, 103 almost complex, 76
Seifert conjecture, 65 K¨hler, 94
a
semisimple, 158, 167 subspace
simple pendulum, 112 coisotropic, 8
simple polytope, 177 isotropic, 5, 8
skew-symmetric bilinear map lagrangian, 8, 46, 77
nondegenerate, 4 symplectic, 5, 8
rank, 4 supercommutativity, 197
standard form, 3 superderivation, 197
symplectic, 4 symplectic
skew-symmetry action, 129
de¬nition, 3 almost symplectic manifold, 74
forms, 13 basis, 5
standard form for bilinear maps, bilinear map, 4
3 blowup, 189
slice theorem, 144 canonical symplectic form on a
smooth polytope, 177 coadjoint orbit, 139, 150,
space 162
a¬ne, 158 cotangent bundle, 9
con¬guration, 107, 113 deformation equivalence, 42
216 INDEX


duality, 5 conservative, 113
equivalence, 42 constrained, 114
equivariant form, 198 mechanical, 113
form, 6, 13
Taubes
Fubini-Study form, 169
CP2 #CP2 #CP2 is not complex,
isotopy, 42
103
linear algebra, 8, 51
unique symplectic structure on
linear group, 72
CP2 , 103
linear symplectic structure, 4
tautological form on T — X
manifold, 6
coordinate de¬nition, 9, 10
measure, 191
intrinsic de¬nition, 10
normal forms, 46
naturality, 11
orthogonal, 8
property, 10
properties of linear symplectic
symplectomorphism, 20
structures, 5
teardrop orbifold, 150
quotient, 141
theorem
reduction, see reduction
Archimedes, 192
strong isotopy, 42
Arnold-Liouville, 110
structure on the space of con-
Atiyah-Guillemin-Sternberg, 136,
nections, 158
170
subspace, 8
Banyaga, 92
toric manifold, see toric mani-
coisotropic embedding, 49
fold
convexity, 170
vector bundle, 74
Darboux, 7, 46, 50
vector ¬eld, 105, 106, 129
Delzant, 136, 179, 189
vector space, 4
Dolbeault, 88
volume, 13, 191, 195
Duistermaat-Heckman, 191, 194
symplectization, 64
µ-neighborhood, 38
symplectomorphic, 5, 42
equivariant coisotropic embed-
symplectomorphism
ding, 193
Arnold conjecture, 33, 55
Euler-Lagrange equations, 123
canonical, 12
Fubini, 195
de¬nition, 7
Gray, 59
equivalence, 15
Hodge, 98“100
exactly homotopic to the iden-
implicit function, 23
tity, 56
local normal form for contact
¬xed point, 33, 55
manifolds, 59
generating function, 23
Marsden-Weinstein-Meyer, 136,
group of symplectomorphisms,
141
12, 53
Moser “ local version, 45
linear, 5
Moser “ version I, 43
recipe, 22
Moser “ version II, 44
tautological form, 20
Newlander-Nirenberg, 82, 88
vs. lagrangian submanifold, 15,
Noether, 147
19
system Picard, 36
217
INDEX


Poincar´ recurrence, 32
e variational
Poincar´™s last geometric theo-
e principle, 113, 114
problem, 113, 122
rem, 33
vector ¬eld
Poincar´-Birkho¬, 33
e
slice, 144 complete, 129
gradient, 107
standard form for skew-symmetric
hamiltonian, 105, 106
bilinear maps, 3
Lie algebra, 164
Stokes, 13, 161
symplectic, 105, 106, 129
symplectomorphism vs. lagrangian
vector space
submanifold, 19
complex, 68
tubular neighborhood, 37, 51
tubular neighborhood in Rn , 41 symplectic, 4
velocity, 119
Weinstein lagrangian neighbor-
volume, 13, 191, 195
hood, 46, 48
Weinstein tubular neighborhood,
weighted projective space, 151
51
Weinstein
Whitehead lemmas, 167
conjecture, 65, 66
Whitney extension, 48, 49
isotropic embedding, 53
Thurston
lagrangian embedding, 51
Kodaira-Thurston example, 102
lagrangian neighborhood theo-
tight contact structure, 66
rem, 46, 48
time-dependent vector ¬eld, 35
Marsden-Weinstein-Meyer quo-
topological constraint, 100
tient, 141
topology of the orbit space, 135
Marsden-Weinstein-Meyer the-
toric manifold
orem, 141
classi¬cation, 177
tubular neighborhood theorem,
de¬nition, 172, 177
51
example, 172
Whitehead lemmas, 167
4-dimensional, 189
Whitney extension theorem, 48, 49
total space, 155
Wirtinger inequality, 117, 118
transitive action, 135
Witten
tubular neighborhood
Seiberg-Witten invariants, 103
equivariant, 143
work, 113
¬bration, 39
homotopy-invariance, 39
Young inequality, 126
in Rn , 41
theorem, 37, 51
Weinstein theorem, 51
twisted product form, 19
twisted projective space, 151

unique symplectic structure on CP2 ,
103
unstable
orbit, 135
point, 112, 152

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