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Akivis M., Goldberg V. — Differential Geometry of Varieties with Degenerate Gauss Maps
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Название: Differential Geometry of Varieties with Degenerate Gauss MapsАвторы: Akivis M., Goldberg V. Аннотация: 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. Both authors have published over 100 papers each. Each has 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2004Количество страниц: 255Добавлена в каталог: 10.06.2008Операции: Положить на полку |
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Предметный указатель
Lightlike hypersurface 154 176 Lightlike line 178 Line of propagation 178 Linear differential form 9 Linear form 6 Linear mapping 11 23 Linear pencil of subspaces 43 Little 47 231 Lobachevsky geometry 27 Lobachevsky space 176 Locally trivial foliation 96 183 Locus of concentration of light rays 191 Locus of condensation of light rays 103 Locus of singular points 46 50 79 Locus of smooth points 50 Lorentzian manifold 176 Lorentzian signature 176 Lyashko xv 103 223 Manifold of singular points 46 Manifold, analytic 5 Manifold, complex 5 Manifold, differentiable 5 46 Manifold, director 61 65 108 150 Manifold, integral 13 Manifold, tangent space to 6 22 Mapping 7 Mapping, bijective 8 Mapping, class of 7 Mapping, differentiable 7 Mapping, Gauss see Gauss map Mapping, geodesic 126 Mapping, Grassmann 42 59 Mapping, injective 8 Mapping, linear 11 23 Mapping, Meusnier — Euler 55 Mapping, surjective 8 Matrix algebra 207 219 Matrix coordinate 210 Matrix representation of algebra 211 Matrix, inverse 16 Matrix, nilpotent 152 Maurer — Cartan equations 12 Metric deformation 87 Metric quadratic form 180 Metric structure 88 134 Metric tensor 176 Metric tensor of de Sitter space 180 Meusnier-Euler mapping 55 Mezzetti xv 88 134 172 231 Minimal surface 88 Minkowski space 184 Misner 177 218 231 Mixed tensor 54 Miyaoka 89 230 Module 9 Monge xii 103 232 Monge — Ampere equation 88 132 133 Monge — Ampere foliation 64—67 72 73 84 85 88 91 92 95 101 118 133 151 152 Monge — Ampere foliation, leaf of 92 93 95—96 101 Mori 134 232 Morimoto xv 88 133 230 232 Moving frame of 214 Moving frame of congruence 196 Moving frame of Grassmannian 59 Moving frame of normalized variety 199 Moving frame of projective plane 28 Moving frame of projective plane over algebra 210 Moving frame of pseudocongruence 198 199 Moving frame of Segre variety 75 Moving frame of variety 51 Multiple component 136 137 142 147 168 Multiple eigenvalue 151 Multiple focus 151—153 Multiple focus hyperplane 151 152 Multiplication of exterior forms 9 Multiplication, exterior 9 n-plane 55 Natural basis 15 16 Natural extension 116 126 127 Natural projection 6 Net of conjugate lines 73 87 171 Net of curvature lines 188 Nilpotent matrix 152 Nirenberg 133 149 172 229 Nomizu xx 5 9 13 19 46 47 88 134 202 230 232 Non-Euclidean space 27 196 199 Non-Euclidean space, elliptic 27 Non-Euclidean space, hyperbolic 126—128 Nondegenerate correlation 80 Nondegenerate mapping 8 Nondegenerate second fundamental form 101 Nondegenerate symmetric affinor 187 Nondegenerate, hyperquadric 27 Nonhomogeneous coordinates 31 106 110 120 Nonhomogeneous parameters 51 Nonsingular matrix 1 2 Nonsingular point 51 Norden 199 202 232 Normal bundle 201 Normal connection 202 218 Normal subspace, first 53 Normal, second 57 Normalization condition 21 23 27 28 Normalized variety 199 203 218 Novikov 96 227 O'Neil 176 232 One-dimensional integral element 14 One-dimensional integral manifold 13 One-dimensional Monge — Ampere foliation 152 One-dimensional subspace 20 One-parameter family of 2-planes 121 162 163 One-parameter family of cones 156 One-parameter family of hyperplanes 73 One-parameter family of plane generators 118 One-parameter family of plane pencils 122 162 163 Operator 4 Opozda 172 232 Orbit 18 Orientable hypersurface 149 Orientation of curve 30 Osculating conic to curve 32—35 Osculating cubic(s) to curve 35—38 Osculating hyperplane of focal line 156 160 Osculating pencil of 36 40 Osculating plane of curve 56 58 67—69 105 160 Osculating plane of dual to third-degree curve 67 Osculating plane of focal line 160 Osculating plane of third-degree curve 67 Osculating subspace of focal surface 164 Osculating subspace of Grassmannian 60 Osculating subspace of hypersurface 61 Osculating subspace of Segre variety 75 Osculating subspace of variety with degenerate Gauss map 86 101 102 129 136 164 170 Oval hyperquadric 177 p-connected manifold 12 p-form(s) 10 p-form(s), closed 12 p-form(s), differential 9 p-form(s), exact 12 p-form(s), exterior 9 p-form(s), exterior, value of 10 Paige 219 232 Pair of coinciding straight lines 80 Pair of intersecting straight lines 80 Parabolic congruence 211 214 Parabolic pencil of hyperspheres 178 184 Parabolic point 88 126 Parabolic surface 127 Parabolic variety 88 126 134 Parabolic variety without singularities 128 132 134 Parabolic variety, complete 126 127 134 Parallel 2-planes 25 Parallel displacement(s) 26 Parallel straight lines 25 Parallel vector field 201 Parameter(s), homogeneous 51 Parameter(s), nonhomogeneous 51 Parameter(s), principal 7 29 Parameter(s), secondary 7 29 30 37 Parametric equations of curve 6 Parametric equations of Veronese variety 82 Parametric variety 92 Pedoe 42 47 229 Pencil of cubics 40 Pencil of matrices 103 104 Pencil of osculating cubics to curve 36 40 Pencil of parallel lines 122 Pencil of second fundamental forms 150 Pencil of straight lines 118 121—123 162 163 169 Pencil of subspaces 43 Pencil of tangent hyperplanes 137 Pfaffian equations, system of completely integrable 13 17 Pfaffian equations, system of in involution 13 46 Pfaffian form 10 Pickert 218 233 Pinkall 172 232 Piontkowski xiv xv 63 72 87—89 133 152 172 228 233 Planar pencil 163 Plane(s) component 137 Plane(s) conisecant 46 Plane(s) element 52 Plane(s), field 43 Plane(s), generator(s) of cone 43 65 74 147 148 Plane(s), generator(s) of cubic symmetroid 46 77 78 Plane(s), generator(s) of cylinder 149 150 Plane(s), generator(s) of dual variety 72 Plane(s), generator(s) of Grassmannian 43 Plane(s), generator(s) of hypersurface with degenerate Gauss map 70 133 Plane(s), generator(s) of natural extension 126 Plane(s), generator(s) of Sacksteder — Bourgain hypersurface 118 Plane(s), generator(s) of Segre variety 44 61 75 Plane(s), generator(s) of variety with degenerate Gauss map 72—74 96 138 167 168 197 Plane(s), pencil of straight lines 164 Plane(s), projective see projective plane Pluecker coordinates 43 Pluecker, hyperquadric 43 Pogorelov 172 233 Pohl 47 231 Poincare lemma 11 Poincare theorem 12 Point(s) at infinity 117 119 Point(s) of condensation 186 Point(s) of projective space 20 Point(s), axial 61 62 Point(s), complex conjugate 86 Point(s), coordinates 23 80 Point(s), homogeneous coordinates of 20 33 Point(s), regular see regular point Point(s), singular see singular point Point(s), smooth see smooth point Point(s), source of light 191 Porteous xiv 233 Positive definite hyperquadric 27 Pottman xiv 229 233 Principal direction of hypersurface 186 Principal form(s) 38 199 Principal parameter(s) 7 29 Product, direct 44 Product, exterior 9 Product, scalar 27 205 Projection of frame vertices 83—84 Projection of space onto subspace 82—83 Projection of Veronese variety 84 85 90 Projective arc length 40 Projective coordinates of point 76 82 84 Projective coordinates of point, homogeneous 41 43 Projective correspondence 123 Projective curvature 40 Projective frame 20 Projective frame, infinitesimal displacement of 21 Projective geometry 96 Projective group 24 Projective group, invariant forms of 24 Projective hyperplane 88 133 Projective plane 19 28 79 Projective plane, infinitesimal displacement of frame of 28 Projective plane, over algebra 207 208 Projective plane, structure equations of 28 Projective realization 126 127 Projective space 19 47 73 96 116 118 198 203 Projective space, dual 70 Projective space, infinitesimal displacement of frame of 21 Projective space, of symmetric matrices 45 81 Projective space, real 126 128 132 133 Projective space, structure equations of 21 Projective structure 88 134 Projective subspace 51 64 65 Projective transformation(s) 20 88 Projectively complete variety 96 150 Projectivization 24 ff. 53 60 Projectivization of osculating subspace 57 86 Projectivization of Segre cone 44 Projectivization of set of symmetric matrices 85 Projectivization of tangent subspace 52 86 Projectivization of vector space 24 Projectivization, basis of 24 Projectivization, center of 24 52 53 60 Projectivization, dimension of 24 Projectivization, infinitesimal displacement of 24 Prolongation, differential 15 Proper Riemannian metric 176 Proper straight line 119 123 Pseudo-Euclidean space 176 208 Pseudo-Riemannian manifold 176 218 Pseudo-Riemannian metric 180 Pseudo-Riemannian space of constant curvature 88 181 Pseudocongruence 196—198 Pseudoelliptic space 176 Pure imaginary plane generator(s) 127 Quadratic hyperband 194 Quadric(s) 65 Quadric(s) imaginary 26 Quartic 46 68 Quasitensor 94 Quotient 19 Quotient group 21 Quotient space 24 r-fold bundle of hyperplanes 141 142 r-fold focus hyperplane 151 r-fold hyperplane 147 151 152 r-fold plane 147 r-multiple eigenvalue 151 r-multiple focus 152 153 Ran 87 233 Rank of Gauss map 63 149 Rank of variety with degenerate Gauss map 63 64 Real affine space 150 Real analytic distribution 15 Real analytic manifold 15 Real cone 108 Real hypercone 112 113 Real hypersurface with degenerate Gauss map 90 Real numbers field 1 Real part of focus hypersurface 127 Real projective space 126 128 132 133 Real rectilinear generator 113 Real singular point 128 132 Real singularity 106 Real straight line 112 Rectilinear congruence 197
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