Differential Geometry of Varieties With Degenerate Gauss Maps

by Akivis, Maks A.; Goldberg, Vladislav V.

• ISBN13: 9780387404639
• ISBN10: 0387404635
• Format: Hardcover
• Copyright: 2003-12-01
• Publisher: Springer Verlag

Summary

In this book the authors study the differential geometry of varieties with degenerate Gauss maps. They use the main methods of differential geometry, namely, the methods of moving frames and exterior differential forms as well as tensor methods. By means of these methods, the authors discover the structure of varieties with degenerate Gauss maps, determine the singular points and singular varieties, find focal images and construct a classification of the varieties with degenerate Gauss maps.The authors introduce the above mentioned methods and apply them to a series of concrete problems arising in the theory of varieties with degenerate Gauss maps. What makes this book unique is the authors' use of a systematic application of methods of projective differential geometry along with methods of the classical algebraic geometry for studying varieties with degenerate Gauss maps.This book is intended for researchers and graduate students interested in projective differential geometry and algebraic geometry and their applications. It can be used as a text for advanced undergraduate and graduate students.Each author has published over 100 papers and they have each written a number of books, including Conformal Differential Geometry and Its Generalizations (Wiley 1996), Projective Differential Geometry of Submanifolds (North-Holland 1993), and Introductory Linear Algebra (Prentice-Hall 1972), which were written by them jointly.

Table of Contents

Preface	p. xi
Foundational Material	p. 1
Vector Space	p. 1
The General Linear Group	p. 1
Vectors and Tensors	p. 3
Differentiable Manifolds	p. 5
The Tangent Space, the Frame Bundle, and Tensor Fields	p. 5
Mappings of Differentiable Manifolds	p. 7
Exterior Algebra, Pfaffian Forms, and the Cartan Lemma	p. 9
The Structure Equations of the General Linear Group	p. 12
The Frobenius Theorem	p. 12
The Cartan Test	p. 13
The Structure Equations of a Differentiable Manifold	p. 15
Affine Connections on a Differentiable Manifold	p. 18
Projective Space	p. 19
Projective Transformations, Projective Frames, and the Structure Equations of a Projective Space	p. 19
The Duality Principle	p. 22
Projectivization	p. 24
Classical Homogeneous Spaces (Affine, Euclidean, Non-Euclidean) and Their Transformations	p. 25
Specializations of Moving Frames	p. 28
The First Specialization	p. 28
Power Series Expansion of an Equation of a Curve	p. 30
The Osculating Conic to a Curve	p. 32
The Second and Third Specializations and Their Geometric Meaning	p. 33
The Osculating Cubic to a Curve	p. 35
Two More Specializations and Their Geometric Meaning	p. 37
Conclusions	p. 39
Some Algebraic Manifolds	p. 41
Grassmannians	p. 41
Determinant Submanifolds	p. 44
Notes	p. 46
Varieties in Projective Spaces and Their Gauss Maps	p. 49
Varieties in a Projective Space	p. 49
Equations of a Variety	p. 49
The Bundle of First-Order Frames Associated with a Variety	p. 51
The Prolongation of Basic Equations	p. 53
The Second Fundamental Tensor and the Second Fundamental Form	p. 54
The Second Fundamental Tensor, the Second Fundamental Form, and the Osculating Subspace of a Variety	p. 54
Further Specialization of Moving Frames and Reduced Normal Subspaces	p. 56
Asymptotic Lines and Asymptotic Cone	p. 58
The Osculating Subspace, the Second Fundamental Form, and the Asymptotic Cone of the Grassmannian	p. 59
Varieties with One-Dimensional Normal Subspaces	p. 61
Rank and Defect of Varieties with Degenerate Gauss Maps	p. 63
Examples of Varieties with Degenerate Gauss Maps	p. 65
Application of the Duality Principle	p. 70
Dual Variety	p. 70
The Main Theorem	p. 72
Cubic Symmetroid	p. 76
Singular Points of the Cubic Symmetroid	p. 78
Correlative Transformations	p. 80
Hypersurface with a Degenerate Gauss Map Associated with a Veronese Variety	p. 81
Veronese Varieties and Varieties with Degenerate Gauss Maps	p. 81
Singular Points	p. 85
Notes	p. 86
Basic Equations of Varieties with Degenerate Gauss Maps	p. 91
The Monge-Ampère Foliation	p. 91
The Monge-Ampère Foliation Associated with a Variety with a Degenerate Gauss Map	p. 91
Basic Equations of Varieties with Degenerate Gauss Maps	p. 92
The Structure of Leaves of the Monge-Ampère Foliation	p. 95
The Generalized Griffiths-Harris Theorem	p. 96
Focal Images	p. 99
The Focus Hypersurfaces	p. 99
The Focus Hypercones	p. 101
Examples	p. 102
The Case n = 2	p. 103
The Case n = 3	p. 104
Some Algebraic Hypersurfaces with Degenerate Gauss Maps in P4	p. 105
The Sacksteder-Bourgain Hypersurface	p. 116
The Sacksteder Hypersurface	p. 116
The Bourgain Hypersurface	p. 118
Local Equivalence of Sacksteder's and Bourgain's Hypersurfaces	p. 123
Computation of the Matrices Ci and B¿ for Sacksteder-Bourgain Hypersurfaces	p. 125
Complete Varieties with Degenerate Gauss Maps in Real Projective and Non-Euclidean Spaces	p. 126
Parabolic Varieties	p. 126
Examples	p. 128
Notes	p. 132
Main Structure Theorems	p. 135
Torsal Varieties	p. 135
Hypersurfaces with Degenerate Gauss Maps	p. 141
Sufficient Condition for a Variety with a Degenerate Gauss Map to be a Hypersurface in a Subspace of PN	p. 141
Focal Images of a Hypersurface with a Degenerate Gauss Map	p. 144
Examples of Hypersurfaces with Degenerate Gauss Maps	p. 145
Cones and Affine Analogue of the Hartman-Nirenberg Cylinder Theorem	p. 146
Structure of Focus Hypersurfaces of Cones	p. 146
Affine Analogue of the Hartman-Nirenberg Cylinder Theorem	p. 149
Varieties with Degenerate Gauss Maps with Multiple Foci and Twisted Cones	p. 151
Basic Equations of a Hypersurface of Rank r with r-Multiple Focus Hyperplanes	p. 151
Hypersurfaces with Degenerate Gauss Maps of Rank r with a One-Dimensional Monge-Ampere Foliation and r-Multiple Foci	p. 152
Hypersurfaces with Degenerate Gauss Maps with Double Foci on Their Rectilinear Generators in the Space P4	p. 154
The Case n = 3 (Continuation)	p. 164
Reducible Varieties with Degenerate Gauss Maps	p. 165
Some Definitions	p. 165
The Structure of Focal Images of Reducible Varieties with Degenerate Gauss Maps	p. 165
The Structure Theorems for Reducible Varieties with Degenerate Gauss Maps	p. 166
Embedding Theorems for Varieties with Degenerate Gauss Maps	p. 169
The Embedding Theorem	p. 169
A Sufficient Condition for a Variety with a Degenerate Gauss Map to be a Cone	p. 172
Notes	p. 172
Further Examples and Applications of the Theory of Varieties with Degenerate Gauss Maps	p. 175
Lightlike Hypersurfaces in the de Sitter Space and Their Focal Properties	p. 176
Lightlike Hypersurfaces and Physics	p. 176
The de Sitter Space	p. 177
Lightlike Hypersurfaces in the de Sitter Space	p. 181
Singular Points of Lightlike Hypersurfaces in the de Sitter Space	p. 184
Lightlike Hypersurfaces of Reduced Rank in the de Sitter Space	p. 192
Induced Connections on Submanifolds	p. 195
Congruences and Pseudocongruences in a Projective Space	p. 195
Normalized Varieties in a Multidimensional Projective Space	p. 198
Normalization of Varieties of Affine and Euclidean Spaces	p. 203
Varieties with Degenerate Gauss Maps Associated with Smooth Lines on Projective Planes over Two-Dimensional Algebras	p. 207
Two-Dimensional Algebras and Their Representations	p. 207
The Projective Planes over the Algebras <$>{\op C}<$>, <$>{\op C}^
<$>, <$>{\op C}^0<$>, and <$>{\op M}<$>	p. 208
Equation of a Straight Line	p. 209
Moving Frames in Projective Planes over Algebras	p. 210
Focal Properties of the Congruences K, K1, and K0	p. 211
Smooth Lines in Projective Planes	p. 214
Singular Points of Varieties Corresponding to Smooth Lines in the Projective Spaces over Two-Dimensional Algebras	p. 215
Curvature of Smooth Lines over Algebras	p. 217
Notes	p. 218
Bibliography	p. 221
Symbols Frequently Used	p. 237
Author Index	p. 239
Subject Index	p. 241
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