model 194 196—199
-triple 303 307
-action 44
—transform 37 206 213
-model 30 176 273
-function 14 50 213 217
—function 50 206 207 215 217
(1, 1)-geodesic map 317
Action group 278—289
Action integral 30
Action Terng 282
Action Uhlenbeck 284
Action, co-adjoint 46
Action, natural 206 211 214 216 282—284
Adler — Kostant — Symes scheme 223—229
Adler — Kostant — Symes scheme and r-matrices 239
Adler — Kostant —Symes scheme in infinite dimensions 231—232
Adler — Kostant —Symes scheme on loop algebras 232—234
Almost complex curve 61 79
Almost complex torus 79
Associate minimal surfaces 42
Associated family 139
Associative 3—planes 315 316
Asymptotic coordinates 17 133
Asymptotic line 111 118 129
Asymptotic line parametrisation 116 117
Automorphism involutive 189
Baecklund transform 17 19 24 25 36 37 41 83 139
Baecklund — Darboux transform 118
Baker function 207 208 216—218
Balanced diagram 96—98
Bianchi diagram 18
Bianchi surfaces 84 116—118 121
Birkhoff decomposition 278
Birkhoff factorisation 217 218
Bogomolny bound 194 198 200
Bonnet surface 84
Bonnet's theorem 134
Borel algebra 304 307
Branch point 32
Bryant correspondence 299 304
Canonical fibration 306
Canonical fibration, associated map of 304 309 310
Cartan immersion 183
Cartan, embedding 41 50 264 284
Casimir 189
Cayley algebra 315
Chebyshev net 115
Chiral 183
Chiral field 149 156 159 163 165 166 169
Chiral model 151 155 159 199
Chiral SU(2)-model 195 196
Clifford torus 76
Cobordant 124
Codazzi — Mainardi equation 36 113
Codazzi — Mainardi — Ricci equations 50
Compatibility condition 36 83 86 88 108 109 113 116 117 120 134 209 211
Complex analytic map 31
Composition law 33
conformal 43
Conformal horizontally 33
Conformal invariance 31 33
Conformal weakly 31—33 277 297 298
Conservation laws 9 10 31 35
Conserved density 9
Constant Gauss curvature 83 84 111—115 129 134 137 138
Constant mean curvature 32 36 37 42 50 60 84 94—101 129 138 139 218
Coxeter — Killing automorphism 215
Critical point 29 31
Current 176
Current left 149 151 160 163 165
Curvature line 93 138
Curvature line coordinates 133
Deformation of a surface 83 84 94 105 106 109 114 117 119 120 122 123
Deformation parameter 94 103 105 109 114
Dirichlet integral 30
Dodd — Bullough equation 24 85
Dressing 84 147 148 153—156 159 170 278—280
Dressing orbit 206
Dressing transform 42
Dual surface 93
Dual symmetry 178
Einstein manifold 301
Ellipse of curvature 77
Elliptic function 35
Energy density 29
Energy integral 29 240
Energy, functional 188 194
Enneper surface 139
Equiharmonic map 70 211 213—215 246
Euler — Lagrange equation 130
Extended framings 257—259
Extended solutions 43 263 274 275
F-structure 213
Factorisation 42 276 283
Factorisation of Lax operator 17
Factorisation problem 148 152—156 161
Finite gap 84 118
Finite type 50 59 68 72 76 79 243
Flag manifold 307
Flag manifolds 298 304
Flag manifolds as k-symmetric spaces 250
Flag manifolds complete 305 310
Flag space 207—208 277
Flag, canonical 282
Flag, manifolds maps into 62
Flag, periodic 212 216
Flatness condition 131
Flip number 89 91 123
Flux 9
force 97
Gauge 39
Gauge group 132
Gauge theory 273
Gauge transformation 25 39
Gauss bundle 277
Gauss equation 36 96 115 116
Gauss map 32 35 37 43 60 92 95 118 133 218
Gauss — Codazzi equations 83 86 94 114 116 119
Gauss — Weingarten equations 36 83 86 93
Geodesic 30
Grassmannian 32 41 207—208
Hamiltonian bi-system 129 137
Hamiltonian system 35 135—138
Hamiltonian theory 47
Hamiltonian vector field 46 47 185
Harmonic 120
Harmonic conjugate 42
Harmonic function 30
Harmonic inverse mean curvature 84
Harmonic maps 29 130 150 160 165 168 169
Harmonic maps and minimal surfaces 32
Harmonic maps between Kaehler manifolds 31
Harmonic maps equivariant 34—35 221
Harmonic maps examples 30—33
Harmonic maps existence problem 34
Harmonic maps from a pseudo-Riemannian manifold 30 117
Harmonic maps from a surface 31 38 41 62
Harmonic maps from a torus 34 35 50 67—68 72 76 79 206 217—218
Harmonic maps from the 2—sphere 31 206
Harmonic maps nonexistence 34
Harmonic maps submersive 32
Harmonic maps to a complex projective space 211—215
Harmonic maps to a flag manifold 206 211—215
Harmonic maps to a homogeneous space 40
Harmonic maps to a Lie group 39—51 260—263 274—284
Harmonic maps to a Lie group of finite type 261—263
Harmonic maps to a Lie group of finite uniton number 267
Harmonic maps to a quaternionic projective space 289
Harmonic maps to a reductive homogeneous space 240—243
| Harmonic maps to a sphere 34 35 50 72 253
Harmonic maps to a symmetric space 50 130 243—245 284—287
Harmonic maps to a symmetric space of finite type 243—245
Harmonic maps to a symmetric space of G-finite type 265—267
Harmonic maps to a unitary group 41—42
Harmonic maps to the 4—sphere 298—300 313 315
Harmonic maps to the 6—sphere 78
Harmonic maps twistorial constructions 32
Harmonic maps, conformal 42 50
Harmonic maps, deformations of 273 287—289
Harmonic maps, framings of 241
Harmonic maps, gauge-theoretic formulation of 242
Harmonic maps, reducible 289
Harmonic maps, spaces of 274 276 287—289
Harmonic maps, weakly conformal 32
Harmonic morphism 33 35
Harmonic sequence 38 61 62 206 213 214
Heat equation 34
Hidden symmetry 273
Higher fundamental form 77
Highest root nilpotent orbit 302 310
Hill's operator 35
Holomorphic map 31 44 50 277
Holomorphic section 37
Holomorphic, quadratic differential 31 37 38
Homogeneous space 177
Homogeneous space, naturally 177
Homogeneous space, reductive 177
Homotopic regularly 123
Homotopy class 34
Hopf problem 138
Hopf, differential 92
Horizontal distribution 298 300 301
Horizontal holomorphic curve 298 300 301 314 315 317
Horizontal holomorphic map 44 55
Inclusive map 301
Infinitesimal action 273
Integrability complete 130 136 187 205
Integrability conditions 39 42 45 63 67 72 131 133 275
Integrability Liouville 183
Integrable 83
Integrable system 273
Isospectral flow 13 16
Isothermal parametrisation 93
Isotopy 123
Isotropic 277 288—289
Isotropic complex 38 63
Isotropic real 50 298
Isotropic strongly 277 289
Isotropy dimension 253
Isotropy order 38 39 64 277
Isotropy, total 39
Iwasawa decomposition 210—212 214 215 218 281
Iwasawa decomposition of a semisimple Lie algebra 228
Iwasawa decomposition of twisted loop algebras 232
Iwasawa decomposition of twisted loop groups 256—257
Jacobi identity, violation of 190
Jacobian 136
Jacobian variety 49 218
K-symmetric space 50 69 239—240
Kac-Moody algebra 205 273
Kac-Moody group 274
Kaehler angle 68
Kaehler manifold 31
KdV equation 8 23 49 206 215
Kink 18
Koszul — Malgrange holomorphic structure 31 37
Lagrangian 195
Laplace's equation 30 33
Laplacian 30 130
Lax equation 187
Lax equations 48 49 137 234
Lax form 187
Lax hierarchy 15
Lax representation 13 83 95 96 115 116 118
Lie algebra 39
Lie algebra, metrisable 177
Lie group 39
Lie sphere geometry 134
Lines of curvature 60
Loop algebras 154 276
Loop algebras, based 260
Loop algebras, twisted 133 232—234
Loop group 161 207 212 215 274 276
Loop group Grassmannian model 280—282
Loop group Grassmannian model of 1—forms 43
Loop group Grassmannian model of connections 137
Loop group Grassmannian model of maps 42—44 133
Loop group Grassmannian model space 44
Lorentz-harmonic maps 117
Maurer — Cartan equations 39 229—231 275
Maurer — Cartan form 39 40 213
Maurer — Cartan form of a Lie group 229
Maurer — Cartan form of a reductive homogeneous space 240
Metric Minkowski 130 133
Metric Sasaki 184
Metric, pseudo-Riemannian 130 133
Minimal branched immersion 32
Minimal fibres 33
Minimal immersion 35
Minimal submanifold 32
Minimal surface 61 84 92—93 129
Miura map 11
MKdV equation 10 23
MKdV hierarchy 17
Moment map 40
Momentum 97
Momentum functional 288
monodromy 89 96—98 121—126 217 218
Monodromy matrices 190
Morse — Bott function 287
Multi-solitons 84
Neumann equation 35
Nilpotent orbits 302
Nilpotent orbits in 308
Nilpotent orbits in 306
Nilpotent orbits, characteristic numbers of 308
Nilpotent orbits, finite covers of 311
Nilpotent orbits, minimal 302 305 310
Nilpotent orbits, regular 302 310
Nilpotent variety 302 309
NLS equation 22
Noether's theorem 40
Nonlinear superposition formula 18
O(3)- -model 194
Octonions 315
One-parameter family of harmonic maps 42
One-parameter family of surfaces 94 101 105 109 111 114 117 119 120
Pants, pair of 123
Parabolic algebra 304 307
Parallel mean curvature 32
Parallel surfaces 98—111
Parity of the spin structure 92 126
Parity of twists 90 91 93
Pauli matrices 87
Pendulum equation 35
Peterson surface 117
Pluriharmonic maps 49 236
Pohlmeyer 182
Poisson bracket 47 185 186
Poisson manifold 47 135
Poisson structure 46 135
Poisson structures 224
Poisson structures from r-matrices 237
polar 75
Polarisation 306
Polynomial flows 21
Polynomial Killing field 138
Primitive maps 61 62 55 68—72 76 213—215 245—248
Primitive maps and twistor lifts 253—256
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