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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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Rectilinear generator of dual variety of curve 73 Rectilinear generator of hypercubic 107 113 122 Rectilinear generator of hypersurface with degenerate Gauss map 73 Rectilinear generator of lightlike hypersurface 183 Rectilinear generator of quadric 65 Rectilinear generator of Sacksteder — Bourgain hypersurface 117 Rectilinear generator of twisted cone 154 Rectilinear generator of variety with degenerate Gauss map 215 Recurrent differential equation 32 Reduced first normal subspace 57 Reducible system of matrices 165 Reducible variety with degenerate Gauss map 165 ff. Regular pencil of second fundamental forms 150 Regular point 50 63 92 93 99 151 Regular point of lightlike hypersurface 186 Regular point of mapping 8 Regular variety 127 Regular variety, complete 118 Relative invariant 5 Relative tensor 4 Relative tensor, law transformation of 5 Relative vector 4 Representation of projective plane over algebra 210 Representation, space of GL(n) 18 Reye 123 233 Ricci tensor 181 Riemannian geometry 96 Riemannian manifold 88 Riemannian space of constant curvature 63 88 134 Riemannian space of constant curvature, elliptic 126—128 Riemannian space of constant curvature, Euclidean see Euclidean space Riemannian space of constant curvature, hyperbolic 126—128 Ring of smooth functions 9 Rogora xv 88 134 172 233 Rosenfeld 23 90 207 210 219 233 234 Roth 45 47 235 Ruled hypersurface 112 Ruled surface 163 Ruled variety 88 Ryzhkov xiv xv 87 223 234 Sacksteder 89 116 134 154 234 Sacksteder — Bourgain hypersurface 116 ff. 134 162 Sacksteder — Bourgain hypersurface, generators of 116 Safaryan 172 234 Sasaki xvii 47 84 172 232 234 Savelyev xiv xv 87 234 Scalar product 27 205 Schafer 219 234 Secant subspace 44 Second fundamental form(s) of focus surface 157 Second fundamental form(s) of Grassmannian 60 Second fundamental form(s) of hypersurface 61 Second fundamental form(s) of normalized variety 199 Second fundamental form(s) of Segre variety 75 Second fundamental form(s) of variety 55—57 61 93 204 Second fundamental form(s) of with degenerate Gauss map 87 98 100 141 142 150—154 Second fundamental tensor of hypersurface 61 141 Second fundamental tensor of variety 54 61 93 199 204 Second normal subspace 57 Second normal(s) 199 Second osculating subspace of Grassmannian 60 Second osculating subspace of variety 56 Second-degree curve 79 Second-order frame(s) 18 22 56 155 Second-order hypercone 108 113 191 Secondary forms 7 30 37 160 Secondary parameters 7 29 37 Segre cone 44 61 87 Segre cone, dimension of 44 Segre cone, director manifold of 61 Segre cone, vertex 61 Segre theorem 63 86 142 171 Segre theorem, generalized 63 86 151 171 Segre variety 44 51 47 61 74 Segre variety, dual defect of 71 Segre variety, plane generator(s) of 61 Segre, C. 47 63 86 87 171 234 235 Semple 45 47 235 Serret 47 235 Severi 47 235 Sextactic point 35 Shafarevich 50 235 Shcherbak xv 235 Simple bundle of hyperplanes 138 140 Simple dyad 44 Simple hyperplane 140 168 Simple plane 138 144 Simple torse 138—140 144 Simply connected curve 28 Simply connected domain 5 126 Simply connected Riemannian space 126 Singular hyperplane 101 Singular point(s) 50 100 183 213 214 Singular point(s) of cubic 35 Singular point(s) of cubic symmetroid 78 79 Singular point(s) of hypercubic 85 106 107 111 Singular point(s) of lightlike hypersurface 146 186 195 Singular point(s) of mapping 8 Singular point(s) of Monge — Ampere foliation 96 Singular point(s) of Sacksteder — Bourgain hypersurface 117 122 Singular point(s) of variety with degenerate Gauss map 215 Singularity 65 88 96 127 Singularity, real 106 Skewed straight line 209 Small dual variety 89 Smooth curve 5 28 66 73 80 Smooth irreducible variety 87 Smooth line 214 219 Smooth point 50 71 Sommese 89 223 Space of constant curvature 63 88 126 Space, affine see affine space Space, cotangent 22 Space, dual see dual space Space, elliptic see elliptic space Space, Euclidean see Euclidean space Space, homogeneous 25 47 90 Space, hyperbolic 126—128 Space, non-Euclidean see non-Euclidean space Space, projective see projective space Space, pseudo-Riemannian 88 Space, Riemannian of constant curvature 87 126 Space, tangent see tangent space Space, vector see vector space Spacelike focal variety 190 Spacelike hyperplane 178 Spacelike hypersurface 176 Spacelike line 178 Spacelike subspace 188 Spacelike variety 188 191 Spatial curve 73 Special linear group 21 Special orthogonal group 181 Specialization of moving frame 28 ff. 159 Sphere 53 Spherical generator 190 191 Spherical map 53 Standard covering 123 Standard leaf 96 Stationary subgroup of planar element 52 Steiner 123 235 Sternberg 133 149 235 Stoker xi xx 172 235 Stokes theorem 12 Straight line 29 Straight line of projective space 20 Strongly parabolic variety 88 Structure equations of affine space 26 203 Structure equations of differentiable manifold 16—17 Structure equations of Euclidean space 27 Structure equations of general linear group 12 18 Structure equations of manifold with affine connection 19 Structure equations of projective plane 28 Structure equations of projective space 21 Structure theorems 169 Strupa 89 231 Subgroup of isotropy transformations 26 Subgroup of parallel displacements 26 Subspace(s) 20 41 65 Subspace(s), basis points of 41 Subspace(s), differential equations of 12 Subspace(s), osculating of focal line 169 Subspace(s), osculating of focal surface 164 Subspace(s), osculating of Grassmannian 60 Subspace(s), osculating of hypersurface 61 Subspace(s), osculating of variety with degenerate Gauss map 86 101 102 129 136 170 171 Subspace(s), projective 64 Subspace(s), tangent see tangent subspace Summation notation 1 Support curve 164 Support submanifold 194 Surface(s), developable 64 87 133 172 Surface(s), minimal 88 Surface(s), parabolic 127 128 Surface(s), ruled 163 Surface(s), Veronese 45 46 79 Surjective mapping 8 Symbol of differentiation with respect to secondary parameters 29 193 Symmetric embedding 45 Symmetric matrix 152 Symmetrization 42 System of matrices, completely reducible 165 System of matrices, irreducible 165 System of matrices, reducible 165 System of Pfaffian equations in involution 13 46 157 158 163 System of Pfaffian equations, completely integrable 13 16 System of second fundamental forms 75 98 101 System of second fundamental tensors 99 System, completely integrable 13 16 Tangent bundle 6 72 73 Tangent bundle of normalized variety 201 Tangent bundle, dimension of 6 Tangent bundle, element of 6 Tangent hypersphere 186 Tangent space 6 15 22 Tangent space, natural projection of 6 Tangent to curve 28 Tangent vector 6 15 Tangent vector, coordinates of 6 15 Tangent, developable 64 Tangent, hyperplane 71 Tangent, hyperplane, to cubic symmetroid 77 78 80 Tangent, hyperplane, to hypersurface with degenerate Gauss map 151 Tangent, subspace to, almost everywhere smooth variety 71 Tangent, subspace to, dual variety 72 Tangent, subspace to, Grassmannian 59 Tangent, subspace to, join 69 Tangent, subspace to, leaf of Monge — Ampere foliation 93 Tangent, subspace to, Segre variety 75 Tangent, subspace to, variety 51 Tangent, subspace to, variety with degenerate Gauss map 63 72 92 93 101 102 Tangential coordinates 23 76 80 101 Tangential coordinates of hyperplane 22 198 200 Tangential frame 23 97 145 Tangential frame, infinitesimal displacement of 23 Tangentially degenerate variety 64 Tangentially nondegenerate hypersurface 104 Tangentially nondegenerate variety 64 65 71—4 81 99 Tensor(s) 4 94 Tensor(s) of normal curvature 202 204 206 Tensor(s), curvature 19 Tensor(s), differential equations of 4 Tensor(s), field 6—7 Tensor(s), inverse 143 Tensor(s), law of transformation of 4 Tensor(s), mixed 54 Tensor(s), relative 4 Tensor(s), relative, differential equations of 4 Tensor(s), second fundamental see second fundamental tensor Tensor(s), torsion 19 Tensorial square 45 Tevelev xvii 74 87 89 236 Theorem, Frobenius 13 17 95 Theorem, Segre 63 86 142 171 Theorem, Segre, generalized 63 86 151 171 Third-degree curve 66 ff. Thorpe 177 218 231 Three-dimensional variety of rank two 128 Timelike hyperplane 178 Timelike hypersurface 176 Timelike line 178 Timelike subspace 187 Timelike variety 187 Tommasi xv 88 134 172 231 Torsal hypersurface 73 123 Torsal variety 135 ff. Torse 62 66—69 74 86 102 104 117 127 128 138—140 144 168 171 Torse, edge of regression of 102 127 188 Torsion tensor of affine connection 19 Torsion-free affine connection 19 Total differential 25 30 40 Transformation(s), admissible 18 52 54 Transformation(s), affine 18 25 Transformation(s), elliptic 27 Transformation(s), Euclidean 27 Transformation(s), identity 20 Transformation(s), projective 20 88 Twisted cone 154 157 Twisted cubic 66 Twisted cylinder 118 154 163 164 Two-dimensional algebra 207 Two-parameter family of hyperplanes 76 Type of variety 87 Typical fiber 52 Unit normal vector 53 Unit tensor 94 Unity point 20 Value of exterior p-form 10 Variety 49 Variety with degenerate Gauss map 64 81 89 Variety with degenerate Gauss map of rank two 67 77 84 164 215 Variety with degenerate Gauss map without singularities 65 127 150 Variety with degenerate Gauss map, basic equations of 94 151 Variety with degenerate Gauss map, completely reducible 165 168 Variety with degenerate Gauss map, irreducible 165 167 168 Variety with degenerate Gauss map, rank of 63 64 Variety with degenerate Gauss map, rectilinear generator of 215 Variety with degenerate Gauss map, reducible 165 Variety with degenerate Gauss map, second fundamental form of 87 98 100 142 150—154 157 Variety with degenerate Gauss map, second osculating subspace of 101 Variety with net of conjugate lines 73 87 Variety, algebraic 42 Variety, complete parabolic 126 127 134 Variety, complete regular 118 Variety, cylindrical 88 Variety, differentiable almost everywhere 49 Variety, moving frame of 51 Variety, of singular points of 100 Variety, parabolic 126 Variety, parabolic, complete 126 127 134 Variety, projectively complete 96 Variety, regular complete 118 Variety, second fundamental form of 55 57 Variety, second fundamental tensor of 54 93 Variety, second osculating subspace of 56 Variety, Segre see Segre variety Variety, strongly parabolic 88 Variety, tangentially degenerate see variety with degenerate Gauss map Variety, tangentially nondegenerate 64 65 81 99 Variety, Veronese see Veronese variety
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