Differential geometry of varieties with degenerate Gauss maps:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York ; Berlin ; Heidelberg ; Hong Kong ; London ; Milan ; Pa
Springer
2004
|
| Series: | CMS books in mathematics
18 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 221 - 236 |
| Physical Description: | XXI, 255 S. graph. Darst. : 24 cm |
| ISBN: | 0387404635 |
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MARC
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| 245 | 1 | 0 | |a Differential geometry of varieties with degenerate Gauss maps |c Maks A. Akivis ; Vladislav V. Goldberg |
| 264 | 1 | |a New York ; Berlin ; Heidelberg ; Hong Kong ; London ; Milan ; Pa |b Springer |c 2004 | |
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| 490 | 1 | |a CMS books in mathematics |v 18 | |
| 500 | |a Literaturverz. S. 221 - 236 | ||
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Record in the Search Index
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| adam_text | Contents
Preface xi
Chapter 1 Foundational Material 1
1.1 Vector Space 1
1.1.1 The General Linear Group 1
1.1.2 Vectors and Tensors 3
1.2 Differentiable Manifolds 5
1.2.1 The Tangent Space, the Frame Bundle, and
Tensor Fields 5
1.2.2 Mappings of Differentiable Manifolds 7
1.2.3 Exterior Algebra, Pfafnan Forms, and the
Cartan Lemma 9
1.2.4 The Structure Equations of the
General Linear Group 12
1.2.5 The Frobenius Theorem 12
1.2.6 The Cartan Test 13
1.2.7 The Structure Equations of a Differentiable Manifold 15
1.2.8 Affine Connections on a Differentiable Manifold 18
1.3 Projective Space 19
1.3.1 Projective Transformations, Projective Frames, and
the Structure Equations of a Projective Space 19
1.3.2 The Duality Principle 22
1.3.3 Projectivization 24
1.3.4 Classical Homogeneous Spaces (Affine, Euclidean,
Non Euclidean) and Their Transformations 25
1.4 Specializations of Moving Frames 28
1.4.1 The First Specialization 28
1.4.2 Power Series Expansion of an Equation of a Curve ... 30
1.4.3 The Osculating Conic to a Curve 32
1.4.4 The Second and Third Specializations and Their
Geometric Meaning 33
1.4.5 The Osculating Cubic to a Curve 35
v
vi Contents
1.4.6 Two More Specializations and Their
Geometric Meaning 37
1.4.7 Conclusions 39
1.5 Some Algebraic Manifolds 41
1.5.1 Grassmannians 41
1.5.2 Determinant Submanifolds 44
Notes 46
Chapter 2 Varieties in Projective Spaces and
Their Gauss Maps 49
2.1 Varieties in a Projective Space 49
2.1.1 Equations of a Variety 49
2.1.2 The Bundle of First Order Frames Associated with
a Variety 51
2.1.3 The Prolongation of Basic Equations 53
2.2 The Second Fundamental Tensor and the Second
Fundamental Form 54
2.2.1 The Second Fundamental Tensor, the Second
Fundamental Form, and the Osculating Subspace
of a Variety 54
2.2.2 Further Specialization of Moving Frames and Reduced
Normal Subspaces 56
2.2.3 Asymptotic Lines and Asymptotic Cone 58
2.2.4 The Osculating Subspace, the Second Fundamental
Form, and the Asymptotic Cone of the Grassmannian 59
2.2.5 Varieties with One Dimensional Normal Subspaces ... 61
2.3 Rank and Defect of Varieties with Degenerate
Gauss Maps 63
2.4 Examples of Varieties with Degenerate Gauss Maps 65
2.5 Application of the Duality Principle 70
2.5.1 Dual Variety 70
2.5.2 The Main Theorem 72
2.5.3 Cubic Symmetroid 76
2.5.4 Singular Points of the Cubic Symmetroid 78
2.5.5 Correlative Transformations 80
2.6 Hypersurface with a Degenerate Gauss Map Associated with
a Veronese Variety 81
2.6.1 Veronese Varieties and Varieties with Degenerate
Gauss Maps 81
2.6.2 Singular Points 85
Notes 86
Contents vii
Chapter 3 Basic Equations of Varieties with
Degenerate Gauss Maps 91
3.1 The Monge Ampere Foliation 91
3.1.1 The Monge Ampere Foliation Associated with
a Variety with a Degenerate Gauss Map 91
3.1.2 Basic Equations of Varieties with Degenerate
Gauss Maps 92
3.1.3 The Structure of Leaves of the
Monge Ampere Foliation 95
3.1.4 The Generalized Griffiths Harris Theorem 96
3.2 Focal Images 99
3.2.1 The Focus Hypersurfaces 99
3.2.2 The Focus Hypercones 101
3.2.3 Examples 102
3.2.4 The Case n = 2 103
3.2.5 The Case n = 3 104
3.3 Some Algebraic Hypersurfaces with Degenerate Gauss Maps
in f4 105
3.4 The Sacksteder Bourgain Hypersurface 116
3.4.1 The Sacksteder Hypersurface 116
3.4.2 The Bourgain Hypersurface 118
3.4.3 Local Equivalence of Sacksteder s and
Bourgain s Hypersurfaces 123
3.4.4 Computation of the Matrices d and Ba for
Sacksteder Bourgain Hypersurfaces 125
3.5 Complete Varieties with Degenerate Gauss Maps
in Real Projective and Non Euclidean Spaces 126
3.5.1 Parabolic Varieties 126
3.5.2 Examples 128
Notes 132
Chapter 4 Main Structure Theorems 135
4.1 Torsal Varieties 135
4.2 Hypersurfaces with Degenerate Gauss Maps 141
4.2.1 Sufficient Condition for a Variety with a Degenerate
Gauss Map to be a Hypersurface in a
Subspace of PN 141
4.2.2 Focal Images of a Hypersurface with a Degenerate
Gauss Map 144
4.2.3 Examples of Hypersurfaces with Degenerate
Gauss Maps 145
viii Contents
4.3 Cones and Affine Analogue of the Hartman Nirenberg
Cylinder Theorem 146
4.3.1 Structure of Focus Hypersurfaces of Cones 146
4.3.2 Affine Analogue of the Hartman Nirenberg
Cylinder Theorem 149
4.4 Varieties with Degenerate Gauss Maps with Multiple Foci 151
and Twisted Cones
4.4.1 Basic Equations of a Hypersurface of Rank r
with r Multiple Focus Hyperplanes 151
4.4.2 Hypersurfaces with Degenerate Gauss Maps of
Rank r with a One Dimensional Monge Ampere
Foliation and r Multiple Foci 152
4.4.3 Hypersurfaces with Degenerate Gauss Maps with
Double Foci on Their Rectilinear Generators in the
Space F4 154
4.4.4 The Case n = 3 (Continuation) 164
4.5 Reducible Varieties with Degenerate Gauss Maps 165
4.5.1 Some Definitions 165
4.5.2 The Structure of Focal Images of Reducible
Varieties with Degenerate Gauss Maps 165
4.5.3 The Structure Theorems for Reducible Varieties
with Degenerate Gauss Maps 166
4.6 Embedding Theorems for Varieties with Degenerate
Gauss Maps 169
4.6.1 The Embedding Theorem 169
4.6.2 A Sufficient Condition for a Variety with a
Degenerate Gauss Map to be a Cone 172
Notes 172
Chapter 5 Further Examples and Applications of
the Theory of Varieties with Degenerate
Gauss Maps 175
5.1 Lightlike Hypersurfaces in the de Sitter Space and Their
Focal Properties 176
5.1.1 Lightlike Hypersurfaces and Physics 176
5.1.2 The de Sitter Space 177
5.1.3 Lightlike Hypersurfaces in the de Sitter Space 181
5.1.4 Singular Points of Lightlike Hypersurfaces in the
de Sitter Space 184
5.1.5 Lightlike Hypersurfaces of Reduced Rank in the
de Sitter Space 192
Contents ix
5.2 Induced Connections on Submanifolds 195
5.2.1 Congruences and Pseudocongruences in a
Projective Space 195
5.2.2 Normalized Varieties in a Multidimensional
Projective Space 198
5.2.3 Normalization of Varieties of Affme and
Euclidean Spaces 203
5.3 Varieties with Degenerate Gauss Maps Associated with
Smooth Lines on Projective Planes over
Two Dimensional Algebras 207
5.3.1 Two Dimensional Algebras and
Their Representations 207
5.3.2 The Projective Planes over the Algebras C,C C°.
and M 208
5.3.3 Equation of a Straight Line 209
5.3.4 Moving Frames in Projective Planes over Algebras ... 210
5.3.5 Focal Properties of the Congruences
K,K andK° 211
5.3.6 Smooth Lines in Projective Planes 214
5.3.7 Singular Points of Varieties Corresponding to Smooth
Lines in the Projective Spaces over
Two Dimensional Algebras 215
5.3.8 Curvature of Smooth Lines over Algebras 217
Notes 218
Bibliography 221
Symbols Frequently Used 237
Author Index 239
Subject Index 241
|
| any_adam_object | 1 |
| author | Akivis, Maks A. 1923- Golʹdberg, Vladislav V. 1936- |
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| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
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| id | DE-604.BV017492087 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:18:41Z |
| institution | BVB |
| isbn | 0387404635 |
| language | English |
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| physical | XXI, 255 S. graph. Darst. : 24 cm |
| publishDate | 2004 |
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| publisher | Springer |
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| series2 | CMS books in mathematics |
| spelling | Akivis, Maks A. 1923- Verfasser (DE-588)128639601 aut Differential geometry of varieties with degenerate Gauss maps Maks A. Akivis ; Vladislav V. Goldberg New York ; Berlin ; Heidelberg ; Hong Kong ; London ; Milan ; Pa Springer 2004 XXI, 255 S. graph. Darst. : 24 cm txt rdacontent n rdamedia nc rdacarrier CMS books in mathematics 18 Literaturverz. S. 221 - 236 Gauss maps Geometry, Differential Gauß-Abbildung (DE-588)4156105-3 gnd rswk-swf Projektive Differentialgeometrie (DE-588)4175883-3 gnd rswk-swf Projektive Differentialgeometrie (DE-588)4175883-3 s Gauß-Abbildung (DE-588)4156105-3 s DE-604 Golʹdberg, Vladislav V. 1936- Verfasser (DE-588)128639644 aut CMS books in mathematics 18 (DE-604)BV013248581 18 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010537753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Akivis, Maks A. 1923- Golʹdberg, Vladislav V. 1936- Differential geometry of varieties with degenerate Gauss maps CMS books in mathematics Gauss maps Geometry, Differential Gauß-Abbildung (DE-588)4156105-3 gnd Projektive Differentialgeometrie (DE-588)4175883-3 gnd |
| subject_GND | (DE-588)4156105-3 (DE-588)4175883-3 |
| title | Differential geometry of varieties with degenerate Gauss maps |
| title_auth | Differential geometry of varieties with degenerate Gauss maps |
| title_exact_search | Differential geometry of varieties with degenerate Gauss maps |
| title_full | Differential geometry of varieties with degenerate Gauss maps Maks A. Akivis ; Vladislav V. Goldberg |
| title_fullStr | Differential geometry of varieties with degenerate Gauss maps Maks A. Akivis ; Vladislav V. Goldberg |
| title_full_unstemmed | Differential geometry of varieties with degenerate Gauss maps Maks A. Akivis ; Vladislav V. Goldberg |
| title_short | Differential geometry of varieties with degenerate Gauss maps |
| title_sort | differential geometry of varieties with degenerate gauss maps |
| topic | Gauss maps Geometry, Differential Gauß-Abbildung (DE-588)4156105-3 gnd Projektive Differentialgeometrie (DE-588)4175883-3 gnd |
| topic_facet | Gauss maps Geometry, Differential Gauß-Abbildung Projektive Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010537753&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013248581 |
| work_keys_str_mv | AT akivismaksa differentialgeometryofvarietieswithdegenerategaussmaps AT golʹdbergvladislavv differentialgeometryofvarietieswithdegenerategaussmaps |