Contemporary Geometry and Related Topics: Proceedings of the Workshop Belgrade, Yugoslavia 15 - 21 May 2002 ( World Scientific )

Contents......Page 8
Preface......Page 6
1. Introduction......Page 10
2. Homogeneous regular   -spaces......Page 12
3. Canonical affinor structures on regular  -spaces......Page 14
4. The algebra A  for regular  -spaces......Page 18
5. The classes of regular  -spaces......Page 20
6. Linear subspaces generated by......Page 22
7. Canonical structures P,J,f,h and Riemannian metrics......Page 24
8. Metric f-structures in generalized Hermitian geometry......Page 26
9. Invariant NKf-structures on naturally reductive homogeneous spaces......Page 27
10. Canonical NKf-structures on regular spaces......Page 29
11.2. The spheres S2, S5, S6......Page 32
11.3. Homogeneous  -spaces of order 8......Page 33
11.4. The 6-dimensional generalized Heisenberg group......Page 34
11.5 The group of hyperbolic motions of the plane R2......Page 36
References......Page 37
1. Introduction......Page 42
2. Preliminaries......Page 43
3. The Main Results......Page 45
References......Page 49
1. Introduction......Page 50
2. Preliminaries......Page 51
3.1. Case when g has a non-trivial center......Page 57
3.2. Case of solvable Lie algebra g and dimg'   2.......Page 60
References......Page 65
A. V. Bolsinov and B. Jovanovic Integrable Geodesic Flows on Riemannian Manifolds: Construction and Obstructions......Page 66
1.1. Geodesics on Riemannian manifolds......Page 67
1.2. Integrable geodesic flows......Page 68
1.3. Statement of the problem......Page 70
2. Classical examples of integrable geodesic flows......Page 71
3.1. Case of two–dimensional surfaces......Page 74
3.2. Topological obstructions in the case of non simply connected manifolds......Page 75
3.3. Topological entropy and integrability of geodesic flows......Page 76
4. Counterexamples......Page 78
5. Geodesic flows on homogeneous spaces and bi-quotients of Lie groups......Page 81
5.1. Non-commutative integrability......Page 82
5.2. Integrable geodesic flows on G / H and K\G/H......Page 84
6.1. Mishchenko–Fomenko conjecture......Page 86
6.2. Integrable pairs......Page 87
7. Integrable deformations of normal metrics......Page 90
8.1. Argument shift method......Page 92
8.2. Chains of subalgebras......Page 94
8.3. Generalized chain method......Page 97
9. Integrability and reduction......Page 99
10. Geodesic flows on the spheres......Page 103
References......Page 107
1. Introduction......Page 114
2. CR submanifolds of maximal CR dimension of complex space forms......Page 116
3. CR submanifolds satisfying h(FX, Y ) + h(X, F Y ) = 0.......Page 118
4. CR submanifolds satisfying h(FX, Y ) - h(X, F Y ) = 0......Page 122
References......Page 124
1. Introduction......Page 126
2.1. Definitions......Page 129
2.2. Motions as curves in SO(n) and o(n )......Page 130
2.3. Moment respect to a geodesic hyperplane and center of mass......Page 132
2.4. The volume of a domain obtained by the motion along a curve......Page 133
2.5. The volume of a hypersurface obtained by the motion along a curve......Page 134
3.1. Definitions......Page 136
3.3. The volume of D......Page 138
3.4. Examples of q-symmetric domains......Page 141
3.5. The volume of C......Page 142
4.1. Definition of holomorphic and Frenet holomorphic motion......Page 144
4.2. The volume of D for holomorphic motions......Page 145
4.3. Bounds for vol(D) when n > 2 and P admits a Frenet motion.......Page 147
4.4. Symmetries on Dp, which make (4.3) true for any motion......Page 148
References......Page 149
1. Introduction......Page 150
2. Non-Archimedean Geometry and p-Adic Numbers......Page 151
3. Adeles and Their Functions......Page 156
4. Quantum Mechanics on Adelic Spaces......Page 158
5. Adelic Quantum Cosmology......Page 160
6. Adelic String/M-theory......Page 163
7. Concluding Remarks......Page 164
References......Page 165
1. Introduction......Page 168
2.1. Classical case......Page 171
2.2. Noncommutative case......Page 173
2.3. Linearization and some examples......Page 175
References......Page 179
1. Introduction......Page 182
2.1. Twisted and warped product structures......Page 183
2.2. Second fundamental form of a map......Page 184
3. Mobius equation associated to a map......Page 185
4. Decomposition results......Page 188
5. Mobius equation and affine maps......Page 192
References......Page 195
1. Introduction......Page 196
2. Preliminaries......Page 198
3. The manifolds M2(n+1)(k)......Page 200
4. Symplectically aspherical manifolds with nontrivial  2 and with no Kahler metrics......Page 204
References......Page 207
1.1. Completely integrable Hamiltonian systems......Page 210
1.2. Liouville's theorem.......Page 211
2. Fomenko-Zieschang invariants of integrable Hamiltonian systems with two degrees of freedom......Page 212
2.1. Isoenergy surfaces......Page 213
2.3. Neighborhoods of singular leaves on the isoenergy surfaces......Page 214
2.4. Gluing matrices......Page 216
3.1. The phase space......Page 218
3.2. Classical cases of integrability: Euler, Lagrange, Kovalevskaya......Page 219
3.3. Liouville classification of the Clebsch case......Page 220
3.4. Marked molecules of the Sokolov case......Page 227
References......Page 231
F. Gavarini The Crystal Duality Principle: From General Symmetries to Geometrical Symmetries......Page 232
1. Introduction......Page 233
2.1. Algebras, coalgebras, and the whole zoo.......Page 237
2.3. Enveloping algebras and symmetric algebras.......Page 239
2.4. Filtrations.......Page 240
3. Connecting functors on (co)augmented (co)algebras......Page 242
4. Connecting and crystal functors on bialgebras and Hopf algebras......Page 246
5.2. Connecting functors and Rees modules.......Page 249
5.3. The bialgebra and Hopf algebra case.......Page 250
6. Deformations II — from Rees bialgebras to quantum groups......Page 252
7. Poisson duality and the Crystal Duality Principle......Page 255
References......Page 258
1. Introduction......Page 260
2. Submanifolds in Riemannian manifold......Page 262
3. Euler-Lagrange equation of the Willmore functional......Page 264
4. Examples of Willmore submanifolds in Sn+P(1)......Page 274
5. Willmore complex submanifolds in a complex space form......Page 277
6. Willmore Lagrangian submanifolds in a complex space form......Page 279
References......Page 282
1. To visualize geometric knowledge or not?......Page 286
2.1. Dynamic geometry environments......Page 287
2.2. Java applets......Page 288
3. Research outcomes and further directions......Page 290
References......Page 291
1. Introduction......Page 294
2. Dual algebraic pairs and polynomial Lie algebras in multiboson physics: a general analysis......Page 296
3. Dual algebraic pairs in action: applications in polarization and nonlinear quantum optics......Page 303
Acknowledgments......Page 307
References......Page 308
1. Introduction and Notations......Page 310
2.2. Line graphics......Page 312
2.3. Independent visibility check......Page 313
3. The Procedure Check......Page 314
4. Contour......Page 315
5 . Curves......Page 317
6. Surfaces......Page 319
6.1. The normal and principal curvature......Page 320
6.2. The geodesic curvature......Page 322
7. Some Animations......Page 325
References......Page 327
1. Introduction......Page 328
2. A method of recognition of conjugate points, based on the distribution of shifts between bounded patterns......Page 330
3. The stability of determination of the projective mapping using a configuration of conjugate points......Page 338
References......Page 341
1. Introduction......Page 344
2. Nondegenerate singularities......Page 347
3. Gauss–Bonnet formula for complex vector bundles......Page 351
4. Some properties of the complex of singularities of integrable Hamiltonian systems......Page 355
References......Page 358
Translators’ Notes:......Page 359
1. Introduction......Page 360
2. Characteristic classes of complex vector bundles......Page 363
3. Characteristic classes of projective modules over an associative algebra......Page 365
4. Mishchenko-Solovjev–Zhuraev construction......Page 373
5. Z/2-graded Lie–Cartan pairs......Page 376
Appendix: The trace of an endomorphism of a projective module......Page 380
References......Page 381
1. Introduction......Page 384
1.1. Some applications......Page 385
1.2. The method of the proof......Page 386
2. Real cohomology structure of closed oriented simply connected four-manifolds......Page 387
3.1. General remarks......Page 388
3.3. Computation of the mnlcs of the homotopy groups......Page 390
4. On homotopy type classification......Page 395
References......Page 397
2.1. Basic notation......Page 398
2.2. Extremal and critical discs of a bounded domain......Page 400
2.3. Levi forms, holomorphic symmetric forms, and normalized defining functions......Page 402
3. Jacobi fields of bounded convex domains......Page 404
4. Busemann functions and horospheres of strongly convex domains......Page 409
5. The eversive map and horospheres......Page 414
References......Page 415
1. Preliminaries......Page 416
2. Canonical normal vectors and normal curvatures of the first order......Page 421
3. Canonical normal vectors and normal curvatures of the second order......Page 426
4. Canonical normal vectors and normal curvatures of the higher order......Page 429
5. Curvature tensors of higher order......Page 430
References......Page 432
1. Previous results......Page 434
2. Commutativity conditions of covariant and Lie derivative, absolute and Lie differential......Page 436
References......Page 439
1. Introduction......Page 440
3. Affine hyperspheres of Type (ii) or (vi)......Page 443
4. Affine hyperspheres of Type (iv)......Page 457
5. Affine hyperspheres of Type (v)......Page 464
References......Page 468