SUBMANIFOLD THEORY: beyond an introduction

Preface......Page 7
Contents......Page 15
 1.1 Gauss and Weingarten Formulas......Page 21
 1.2 Interpretations of the Second Fundamental Form......Page 24
  1.2.1 The Second Fundamental Form in Euclidean Space......Page 26
  1.2.2 The Gauss Map of a Euclidean Hypersurface......Page 29
 1.3 The Gauss, Codazzi and Ricci Equations......Page 32
  1.3.1 The Fundamental Theorem of Submanifolds......Page 35
 1.4 The Basic Equations of a Hypersurface......Page 36
  1.4.2 The Principal Curvatures......Page 37
  1.4.3 Holonomic Hypersurfaces......Page 39
 1.5 Totally Geodesic Submanifolds......Page 41
 1.6 The Relative Nullity Distribution......Page 44
 1.7 Umbilical Submanifolds......Page 45
 1.8 Principal Normals......Page 49
 1.9 Submanifolds with Flat Normal Bundle......Page 54
 1.10 Proof of the Fundamental Theorem......Page 56
 1.11 Appendix: Burstin–Mayer–Allendoerfer Theory......Page 61
 1.12 Notes......Page 62
 1.13 Exercises......Page 65
 2.1 Basic Facts......Page 82
  2.2.1 The Main Result......Page 85
  2.2.2 The s-Nullities......Page 88
  2.2.3 The Type Number......Page 89
 2.3 An Application......Page 90
 2.5 Exercises......Page 99
3 Minimal Submanifolds......Page 104
 3.1 The First Variational Formula......Page 105
 3.2 Euclidean Minimal Submanifolds......Page 107
 3.3 Minimal Submanifolds of the Sphere......Page 108
  3.3.1 Standard Minimal Immersions of Spheresinto Spheres......Page 109
 3.4 The Ricci Tensor of a Submanifold......Page 112
 3.5 Rigidity of Minimal Hypersurfaces......Page 114
 3.6 The Ricci Condition......Page 117
 3.7 Notes......Page 120
 3.8 Exercises......Page 122
4 Local Rigidity of Submanifolds......Page 128
 4.1 Flat Bilinear Forms......Page 129
  4.1.1 Indefinite Inner Products......Page 130
  4.1.2 Basic Properties of Flat Bilinear Forms......Page 132
  4.1.3 The Chern–Kuiper Inequalities......Page 135
  4.1.4 The Beez-Killing Theorem......Page 136
 4.2 Uniqueness of the Normal Connection......Page 138
 4.3 The Allendoerfer Rigidity Result......Page 140
  4.4.1 The Main Lemma......Page 143
  4.4.2 The do Carmo–Dajczer Rigidity Result......Page 144
  4.4.3 A Counterexample......Page 145
 4.5 Appendix: Proof of the Main Lemma......Page 147
 4.6 Notes......Page 151
 4.7 Exercises......Page 153
5 Constant Curvature Submanifolds......Page 157
 5.1 The Structure of the Second Fundamental Form......Page 158
 5.2 Principal Coordinates......Page 163
 5.3 The Associated Systems of PDEs......Page 166
 5.4 The Case of Distinct Curvatures......Page 169
 5.5 Complete Hyperbolic Submanifolds......Page 177
 5.6 Notes......Page 179
 5.7 Exercises......Page 181
6 Submanifolds with Nonpositive Extrinsic Curvature......Page 187
 6.1 Otsuki's Lemma......Page 188
 6.2 Cylindrically Bounded Submanifolds......Page 191
 6.3 Florit's Nullity Estimate......Page 196
 6.4 Notes......Page 199
 6.5 Exercises......Page 200
7 Submanifolds with Relative Nullity......Page 202
 7.1 The Splitting Tensor......Page 203
  7.1.1 The Splitting Tensor of the Relative NullityDistribution......Page 204
  7.1.2 Submanifolds with Umbilical Conullity......Page 206
 7.2 Completeness of the Relative Nullity Foliation......Page 208
  7.2.1 The Case of Pairs of Immersions......Page 210
  7.2.2 The Spherical Case......Page 211
  7.2.3 The Euclidean Case......Page 213
 7.3 The Gauss Parametrization......Page 215
  7.3.1 Some Applications......Page 224
 7.4 Intrinsically Homogeneous Hypersurfaces......Page 227
 7.5 Notes......Page 231
 7.6 Exercises......Page 233
 8.1 Product Manifolds......Page 238
 8.2 Extrinsic Products of Immersions......Page 240
 8.3 The Basic Decomposition Theorems......Page 242
  8.4.1 Curvature Conditions......Page 250
  8.4.2 Conditions on the s-Nullities......Page 254
 8.5 Global Conditions for Decomposability......Page 257
 8.6 Notes......Page 263
 8.7 Exercises......Page 264
9 Conformal Immersions......Page 269
 9.1 The Paraboloid Model......Page 270
 9.2 The Space of Euclidean Hyperspheres......Page 272
 9.3 Envelopes of Congruences of Hyperspheres......Page 274
  9.3.1 The Conformal Gauss Parametrization......Page 279
  9.3.2 Envelopes and Dupin Principal Normal VectorFields......Page 282
 9.4 The Light-Cone Representative......Page 284
 9.5 Rigidity of the Paraboloid Model......Page 289
 9.6 The Second Fundamental Form of the Light-Cone Representative......Page 291
 9.7 Conformal Congruence of Submanifolds......Page 294
 9.8 A Fundamental Theorem in Moebius Geometry......Page 296
 9.9 Conformal Rigidity of Euclidean Submanifolds......Page 306
 9.10 Conformal Immersions of Products......Page 309
 9.11 Notes......Page 317
 9.12 Exercises......Page 320
 10.1 Polar Metrics on Product Manifolds......Page 324
  10.2.1 Partial Tubes in Euclidean Space......Page 326
  10.2.2 Partial Tubes in the Sphere and the Hyperbolic Space......Page 332
  10.2.3 Partial Tubes Over Extrinsic Products......Page 337
  10.2.4 The Decomposition Theorem......Page 339
  10.3.1 Warped Products of Immersions into Euclidean Space......Page 343
  10.3.2 Warped Products of Immersions into the Hyperbolic Space......Page 349
  10.3.3 Nölker's Theorem......Page 357
 10.4 Immersions of Warped Products and s-Nullities......Page 361
 10.5 Notes......Page 366
 10.6 Exercises......Page 367
11 The Sbrana–Cartan Hypersurfaces......Page 372
  11.1.1 Hyperbolic, Parabolic and Elliptic Hypersurfaces......Page 373
  11.1.2 Projectable Vector Fields and Tensors......Page 380
  11.1.3 Hyperbolic and Elliptic Surfaces......Page 383
  11.1.4 From Hypersurfaces to Surfaces and Backwards......Page 386
 11.2 Surfaces of First and Second Species......Page 389
  11.2.1 Surfaces of First and Second Species of Real Type......Page 392
  11.2.2 Surfaces of First and Second Species of ComplexType......Page 393
 11.3 The Parametric Description......Page 394
 11.4 Notes......Page 396
 11.5 Exercises......Page 397
12 Genuine Deformations......Page 401
 12.1 Ruled Extensions......Page 403
 12.2 Pairs of Ruled Extensions......Page 408
  12.2.1 Constructing Pairs of Ruled Extensions......Page 413
 12.3 Genuine Isometric Deformations......Page 417
 12.4 Genuine Rigidity......Page 428
 12.5 Compositions......Page 429
 12.6 Submanifolds of Two Space Forms......Page 431
  12.7.1 Reduction to the Isometric Case......Page 439
  12.7.2 Conformally Ruled Extensions of Conformal Pairs......Page 441
  12.7.4 Compositions of Conformal Immersions......Page 449
  12.7.5 Genuine Conformal Pairs from Isometric Ones......Page 451
 12.8 Singular Genuine Deformations......Page 453
 12.9 Nonparallel First Normal Bundle......Page 457
 12.10 Notes......Page 462
 12.11 Exercises......Page 464
13 Deformations of Complete Submanifolds......Page 467
 13.1 The Sacksteder Theorem......Page 468
 13.2 Ruled Submanifolds......Page 471
 13.3 Submanifolds of Rank at Most Two......Page 473
 13.4 The Hypersurface Case......Page 477
 13.5 The Compact Case......Page 481
 13.6 Notes......Page 485
 13.7 Exercises......Page 486
14 Infinitesimal Bendings......Page 490
 14.1 Infinitesimal Bendings......Page 491
 14.2 Infinitesimal Rigidity......Page 492
  14.3.1 The Integrability Conditions for an Infinitesimal Bending......Page 495
  14.3.3 Special Hyperbolic and Elliptic Surfaces......Page 502
  14.3.4 The Classification......Page 503
  14.3.5 Infinitesimally Bendable Hypersurfacesas Envelopes......Page 505
 14.4 Sbrana–Cartan Hypersurfaces as Envelopes......Page 509
 14.6 Exercises......Page 511
15 Real Kaehler Submanifolds......Page 513
 15.1 Some Basic Facts......Page 514
 15.2 Unboundedness of Real Kaehler Submanifolds......Page 516
 15.3 Minimal Real Kaehler Submanifolds......Page 518
  15.3.1 The Associated Family......Page 519
 15.4 Real Kaehler Hypersurfaces......Page 524
 15.5 The Codimension Two Case......Page 528
 15.6 The Case of Nonflat Ambient Spaces......Page 534
 15.7 Hypersurfaces Making a Constant Angle......Page 537
 15.8 Appendix......Page 545
 15.9 Notes......Page 547
 15.10 Exercises......Page 549
16 Conformally Flat Submanifolds......Page 553
 16.1 Hypersurfaces of the Light-Cone......Page 554
 16.2 Conformally Flat Manifolds......Page 557
  16.3.1 Structure of the Second Fundamental Form......Page 559
  16.3.2 A Nonparametric Description......Page 563
 16.4 A Parametrization for Codimension Two......Page 565
 16.5 Conformally Flat Hypersurfaces of Dimension Three......Page 567
 16.6 Notes......Page 573
 16.7 Exercises......Page 576
 17.1 Cartan Hypersurfaces......Page 580
 17.2 The First Step......Page 584
 17.3 The Reduction......Page 602
 17.4 The Classification......Page 606
 17.5 Notes......Page 607
 17.6 Exercises......Page 609
Appendix A Vector Bundles......Page 610
Bibliography......Page 621
Index......Page 634