Projective Differential Geometry of Submanifolds (eBook)

Front Cover 1
Projective Differential Geometry of Submanifolds 4
Copyright Page 5
Table of Contents 10
Preface 6
Chapter 1. Preliminaries 14
1.1 Vector Spaces 14
1.2 Differentiable Manifolds 18
1.3 Projective Space 30
1.4 Some Algebraic Manifolds 39
Notes 44
Chapter 2. The Foundations of Projective Differential Geometry of Submanifolds 46
2.1 Submanifolds in a Projective Space and Their Tangent Subspaces 46
2.2 The Second Fundamental Form of a Submanifold 51
2.3 Osculating Subspaces and Fundamental Forms of Higher Orders of a Submanifold 55
2.4 Asymptotic and Conjugate Directions of Different Orders on a Submanifold 60
2.5 Some Particular Cases and Examples 66
2.6 Classification of Points of Submanifolds by Means of the Second Fundamental Form 74
Notes 83
Chapter 3. Submanifolds Carrying a Net of Conjugate Lines 86
3.1 Basic Equations and General Properties 86
3.2 The Holonomicity of the Conjugate Net S2 89
3.3 Classification of Conjuagate Nets S2 94
3.4 Some Existence Theorems 98
3.5 The Laplace Transfor Generalizations 103
3.6 Conic m-Conjugate System 116
Notes 124
Chapter 4. Tangentially Degenerate Submanifolds 126
4.1 Basic Notions and Equations 126
4.2 Focal Images 130
4.3 Decomposition of Focal Images 133
4.4 The Holonomicity of the Focal Net 135
4.5 Some Other Classes of Tangentially Degenerate Submanifolds 139
4.6 Manifolds of Hypercones 143
4.7 Parabolic Submanifolds without Singularities in Euclidean and Non-Euclidean Spaces 145
Notes 154
Chapter 5. Submanifolds with Asymptotic and Conjugate Distributions 156
5.1 Distributions on Submanifolds of a Projective Space 156
5.2 Asymptotic Distributions on Submanifolds 158
5.3 Submanifolds with a Complete System of Asymptotic Distributions 161
5.4 Three-Dimensional Submanifolds Carrying a Net of Asymptotic Lines 164
5.5 Submanifolds with a Complete System of Conjugate Distributions 178
Notes 184
Chapter 6. Normalized Submanifolds in a Projective Space 186
6.1 The Problem of Normalization of a Submanifold in a Projective Space 186
6.2 The Affine Connection on a Normalized Submanifold 191
6.3 The Connection in the Normal Bundle 195
6.4 Submanifolds with a Flat Normal Connection 201
6.5 Intrinsic Normalization of Submanifolds 205
6.6 Normalization of Submanifolds Carrying a Conjugate Net of Lines 212
Notes 218
Chapter 7. Projective Differential Geometry of Hypersurfaces 222
7.1 Basic Equations of the Theory of Hypersurfaces 222
7.2 Osculating Hyperquadrics of a Hypersurface 229
7.3 Invariant Normalizations of a Hypersurface 235
7.4 The Rigidity Problem in a Projective Space 247
7.5 The Geometry of a Surface in Three-Dimensional Projective Space 258
7.6 The Geometry of Hyperbands 268
Notes 279
Chapter 8. Algebraization Problems in Projective Differential Geometry 282
8.1 The First Generalization of Reiss’ Theorem 284
8.2 The Second Generalization of Reiss’ Theorem 290
8.3 Degenerate Monge’s Varieties 292
8.4 Submanifolds with Degenerate Bisecant Varieties 298
Notes 308
Bibliography 310
Symbols Frequently Used 346
Index 348