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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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Enlarged affene space 119 Envelope 62—71 73 76 84 85 120 Equation(s) of asymptotic cone 59—61 Equation(s) of asymptotic lines 58 Equation(s), cubic 35 Equation(s), focus hypercone 101 Equation(s), focus hypersurface 100 Equation(s), hyperplane 22 Equation(s), osculating, conic 33 34 Equation(s), osculating, cubic 38 Equation(s), pencil of osculating cubics 36 Equation(s), Segre variety 44 Equation(s), straight line 209 Equation(s), tangent, subspace to Segre variety 75 Equation(s), tangent, to curve 34 Equivalence classes 24 Equivalence relation 19 24 53 Euclidean geometry 96 Euclidean space 26 27 47 53 58 64 87 88 126—128 133 134 149 196 198 199 205 218 Euclidean space, structure equations of 27 Euclidean transformations 27 Euler theorem 85 Event horizon 176 Exact p-form 12 Exterior algebra 10 Exterior differentiation 11 12 Exterior differentiation of product 11 Exterior multiplication 9 Exterior p-form(s) 9 Exterior product 9 Exterior quadratic form 9 10 Factorization of 24 Family of hypercones 146 Family of hyperplanes 70 Family of moving frames of Grassmannian 59 Family of moving frames of hypercone 145 Family of moving frames of variety 51 Family of parallel varieties 205 Family of rectilinear generators 183 Family of second-order cones 188 190 191 Family of torses 73 Fania 89 223 Fiber cone 154 162 Fiber forms 7 Fiber forms of dual variety 98 Fiber forms of first-order frame bundle 17 52 Fiber forms of second-order frame bundle 17 Fiber parameters 7 Fibration 6 Field of complex numbers 1 150 Field of real numbers 1 150 Finikov 15 38 40 46 47 103 196 227 First integral(s) 16 92 First normal subspace 53 First normal(s) 199 First-order frames 22 51 Fischer xiv 63 87—89 133 134 228 Flat affine connection 202 205 Flat normal connection 203 218 Focal image 101 135 141 144 165 169 Focal line 155 159 162 Focal net 69 Focal surface 103 155 Focal variety 197 Focal variety of lightlike hypersurface 186 187 Focus 87 104 105 117 127 128 152—55 191 196 200 Focus of rectilinear generator 73 102 Focus, double 117 154 157 164 213 Focus, hypercone 101 136—142 144 147 166 169 198 Focus, hypercone, of pseudocongruence 200 201 204 Focus, hyperplane 101 198 Focus, hypersurface 100 110 115 136—138 144 151 165 166 168 197 Focus, hypersurface, of cone 102 147—148 Focus, hypersurface, of congruence 200 204 205 Focus, hypersurface, of cubic symmetroid 103 Focus, hypersurface, of hypercubic 110 115 Focus, hypersurface, of hypersurface 103 Focus, hypersurface, of join 102 Focus, hypersurface, of reducible variety 165—166 Focus, hypersurface, of torse 102 Focus, hypersurface, real part of 127 Focus, point 127 fold 188 Foliation(s) 8 96 183 Fomenko 96 227 Form(s), basis see basis forms Form(s), closed 5 12 Form(s), connection 19 Form(s), exact 12 Form(s), exterior 9 Form(s), fiber 7 17 52 97 Form(s), invariant of general linear group 18 Form(s), linear differential 9 Form(s), Pfaffian 10 Form(s), principal 38 192 199 Form(s), second fundamental see second fundamental form Form(s), secondary 7 30 37 Fourth-degree curve 46 68 69 Frame(s) 1 Frame(s) bundle 6 Frame(s) bundle of second-order frames 17 Frame(s) bundle, dimension of 6 Frame(s) of first order 18 22 Frame(s) of second order 18 22 Free module 9 frenet 47 228 Frenet formulas 40 Frobenius theorem 12 13 16 95 Fundamental group of de Sitter space 180 Gardner xx 15 46 216 225 Gauss xi 228 Gauss defect 63 64 71 72 126 134 Gauss map 53 63 91 133 149 172 Gaussian curvature 149 150 Gaussian torsion 202 206 General linear group 1 12 26 General linear group, invariant forms of 2 7 18 General linear group, representation space of 18 General linear group, structure equations of 12 18 General relativity 176 Generalized Griffiths — Harris theorem 93 97 99 Generalized Segre theorem 63 86 151 Generator(s) of cone 65 74 Generator(s) of cubic symmetroid 46 Generator(s) of cylinder 149 150 Generator(s) of free module 9 Generator(s) of hypersurface 146 Generator(s) of light cone 178 Generator(s) of Segre variety 44 45 Generator(s) of variety with degenerate Gauss map 72—74 96 Geodesic completeness 96 Geodesic mapping 126 Geometric optic 186 Geometry, algebraic 87 88 Geometry, Euclidean 96 Geometry, hyperbolic 27 176 Geometry, projective 96 Geometry, Riemannian 96 Giblin xiv 224 Goldberg xiv xvi xviii xx 15 46 86 89 101 133 134 149 154 169 171—173 176 178 184 186 189 194 199 218 219 222 223 Goldsmith xx 15 46 216 225 Goryunov xv 103 223 Goursat 132 228 Grassmann coordinates 41 42 47 Grassmann mapping 42 59 Grassmannian 41—43 47 63 86 209 219 Griffiths xiv xvii xx 15 22 46 47 74 86—89 91 99 133 134 216 225 228 229 Griffiths — Harris theorem 99 Griffiths — Harris theorem, generalized 93 97 99 Group of admissible transformations 18 Group of admissible transformations of coframes 180 Group of affine transformations 18 26 Group of motions 27 176 Group of projective transformations 20 Group of rotations 176 Group quotient 21 Group, cyclic 21 Group, general linear 1 12 26 Halphen 40 229 Halphen's point 40 41 Harris xiv xvii 22 47 55 74 86—89 99 133 134 172 227 228 Hartman 133 149 172 229 Hartman — Nirenberg cylinder theorem 149—50 172 Hartman — Nirenberg cylinder theorem, affine analog of 150 Hodge 42 47 229 Homogeneous coordinates of hyperplane 22 Homogeneous coordinates of point 20 33 35 41 43 45 50 67 94 106 110 116 119 Homogeneous function 50 Homogeneous parameters 51 Homogeneous space 25 47 90 181 Horizon 176 Horizon, Cauchy 176 Horizon, event 176 Horizon, Kruskal 176 Horizontal distribution 18 Horizontal forms 97 98 Hoschek xiv 229 Hyperbolic congruence 211 213 Hyperbolic geometry 27 176 Hyperbolic pencil of hyperspheres 178 Hyperbolic space 126—128 Hypercone(s) 81 146 Hypercone(s) of second order 108 Hypercone(s), complex conjugate 113 Hypercone(s), focus 101 Hypercone(s), real 112 113 Hypercubic 46 106 110 134 Hyperplane(s) 22 56 61 139 145 147—49 151 152 154 164 177 196 198 200 Hyperplane(s) at infinity 25 118 119 126 150 163 164 172 203—206 Hyperplane(s), equation of 22 Hyperquadric 145 Hyperquadric convex 126 Hyperquadric imaginary 126 Hyperquadric invariant 27 Hyperquadric nondegenerate 27 Hyperquadric Pluecker 43 Hyperquadric positive definite 27 Hypersphere 177 191 Hypersurface(s) 8 61 149 169 172 Hypersurface(s) with degenerate Gauss map 70—74 81 84 85 103 107 122 141 151 172 183 Hypersurface(s) with degenerate Gauss map of rank two 107 113 134 156 Hypersurface(s) with degenerate Gauss map, algebraic 90 Hypersurface(s) with degenerate Gauss map, real 113 Hypersurface(s) with degenerate Gauss map, singular point of 117 Hypersurface(s), cubic 90 Hypersurface(s), focus 100 Hypersurface(s), level 8 Hypersurface(s), ruled 112 Hypersurface(s), tangentially nondegenerate 64 Ideal hyperplane 25 26 Identity matrix 94 Identity transformation 20 Image of Grassmannian 42 43 51 59 Imaginary focus hypersurface 128 129 Imaginary, hyperquadric 126 Imaginary, quadric 26 Improper hyperplane 25 116 126 incidence 23 70 Index of relative nullity 63 88 126 134 Infinitesimal displacement of frame 2 98 Infinitesimal displacement of frame of affine space 25 203 Infinitesimal displacement of frame of congruence 196 Infinitesimal displacement of frame of de Sitter space 179 Infinitesimal displacement of frame of n-plane 55 Infinitesimal displacement of frame of normalized variety 199 Infinitesimal displacement of frame of projective hyperplane 98 Infinitesimal displacement of frame of projective plane 28 211 Infinitesimal displacement of frame of projective space 21 24 Infinitesimal displacement of frame of projectivization 24 Infinitesimal displacement of frame of variety with degenerate Gauss map 93 Infinitesimal displacement of frame of vector space 2 Infinitesimal displacement of frame, tangential 23 97 Injective mapping 8 Integral element 14 Integral element, one-dimensional 14 Integral element, three-dimensional 14 Integral element, two-dimensional 14 Integral first 16 92 Integral manifold 13 Intersecting straight lines 209 Invariance of exterior differential 12 Invariant absolute 5 Invariant distribution 19 Invariant forms of general linear group 2 7 18 Invariant forms of group 18 Invariant horizontal distribution 18 Invariant projective group 24 Invariant subgroup 26 Invariant, hyperquadric 27 Invariant, relative 5 Inverse matrix 3 16 Inverse tensor 143 Invertible function 5 Invertible transformation 5 Involutive automorphism 90 Irreducible system of matrices 165 Irreducible variety 71 73 165 167 168 Ishikawa xv 88—90 92 118 119 133 154 172 219 229 230 Isotropic cone 176 177 Isotropic cone rectilinear generator of 178 Isotropy transformation 26 Jacobi matrix 8 50 99 186 Jacobian 99 101 195 Jensen 15 46 229 Jijtchenko 47 230 Join 69 102 105 107 113 Kimura 89 230 Knot 36 38 Kobayashi xx 5 9 13 19 46 47 88 202 226 230 Konnov 47 230 Kossowski 184 230 Kronecker symbol 27 Kruskal horizon 176 Kuiper 63 88 134 226 Kupeli 218 230 l-cylinder 149 150 l-parabolic varieties 126 127 Landsberg xiv xvi xvii 86 87 89 133 231 Lanteri 89 231 Laplace transform 134 Laptev 17—19 47 231 Lashof 133 149 226 Law of transformation of covector 4 Law of transformation of relative tensor 5 Law of transformation of tensor 4 Law of transformation of vector 3 Leaf of cubic symmetroid 77 Leaf of Gauss map 64 92 Leaf of Monge — Ampere foliation 64 92 93 95—96 101 Left-invariant form(s) 2 Level hypersurfaces 8 Lichnerowicz 19 47 231 Lie algebra 19 Light cone 176 Light cone of de Sitter space 178 191 Light flux 184 191 Light impulse 176 178 184 Light ray 103 186 Lighting variety 191 Lightlike hyperplane 176 ff.
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