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M.A.Akivis, V.V.Goldberg — Projective Differential Geometry of Submanifolds
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Название: Projective Differential Geometry of SubmanifoldsАвторы: M.A.Akivis, V.V.Goldberg Аннотация: In this book, the general theory of submanifolds in a multidimensional projective space is constructed. The topics dealt with include osculating spaces and fundamental forms of different orders, asymptotic and conjugate lines, submanifolds on the Grassmannians, different aspects of the normalization problems for submanifolds (with special emphasis given to a connection in the normal bundle) and the problem of algebraizability for different kinds of submanifolds, the geometry of hypersurfaces and hyperbands, etc. A series of special types of submanifolds with special projective structures are studied: submanifolds carrying a net of conjugate lines (in particular, conjugate systems), tangentially degenerate submanifolds, submanifolds with asymptotic and conjugate distributions etc. The method of moving frames and the apparatus of exterior differential forms are systematically used in the book and the results presented can be applied to the problems dealing with the linear subspaces or their generalizations.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1993Количество страниц: 362Добавлена в каталог: 11.12.2009Операции: Положить на полку |
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Hypersurface(s), conjugate directions of 54 81 211 Hypersurface(s), cubic 286 Hypersurface(s), discriminant of 226 Hypersurface(s), discriminant of 2nd order of 210 224 246 Hypersurface(s), dual image of 210 Hypersurface(s), dual image of, 2nd fundamental form of 211 Hypersurface(s), dual image of, generating element of 210 Hypersurface(s), Gauss image of 210 211 Hypersurface(s), intrinsic normalization of 222 224 Hypersurface(s), invariant normalization of 222 267 Hypersurface(s), moving frame of 209 Hypersurface(s), normalizing object of 224 227 Hypersurface(s), osculating subspace of 54 Hypersurface(s), pair of 272 Hypersurface(s), pair of, moving frame of 272 Hypersurface(s), pencil of Darboux hyperquadrics of 218 223 230 231 235 237 249 267 Hypersurface(s), plane generator of 131 Hypersurface(s), projectively equivalent 234 240 254 Hypersurface(s), ruled 171 Hypersurface(s), system of 271 Hypersurface(s), tangent hyperplane to 210 216 272 Hypersurface(s), tangentially degenerate 256 260 Hypersurface(s), tangentially degenerate of rank one 116 Hypersurface(s), tangentially degenerate of rank two 291 Hypersurface(s), tangentially nondegenerate 222 Hypersurface(s), with parallel 2nd fundamental form vi 221 222 Ideal hyperplane 21 Identity, matrix 82 226 Identity, Ricci 179 Identity, transformation 1 18 Image of Gauss mapping 38 56 59 113 Image of Grassmannian 27 28 34 50 Image of Grassmannian, 2nd fundamental form of 51 Image of Grassmannian, 2nd osculating subspace of 51 Image of Grassmannian, asymptotic cone of 51 Image of Grassmannian, asymptotic cone of order k of 53 Image of Grassmannian, asymptotic direction of order k of 53 Image of Grassmannian, dimension of 35 36 Image of Grassmannian, fundamental form of order k of 52 Image of Grassmannian, osculating subspace of order k of 52 Image of Grassmannian, plane generator of 28 Image of Grassmannian, tangent subspace to 35 Image, focal 121—123 129 Image, focal, dual 119 Imaginary, focus variety 137 Imaginary, hyperquadric 134 Imaginary, quadric 22 Improper, hyperplane 134 136 Improper, plane 137 incidence 20 116 265 Index notations vi Index of relative nullity 133 141 Index of relative nullity, constant 133 134 Infinitesimal displacement of frame of 2-submanifold 287 Infinitesimal displacement of frame of affine space 21 Infinitesimal displacement of frame of congruence 128 Infinitesimal displacement of frame of conjugate net 103 Infinitesimal displacement of frame of curve 53 Infinitesimal displacement of frame of dual space 131 Infinitesimal displacement of frame of hyperband 257 Infinitesimal displacement of frame of hypercone 283 Infinitesimal displacement of frame of projective space 19—20 287 Infinitesimal displacement of frame of pseudocongruence 129 Infinitesimal displacement of frame of submanifold 74 Infinitesimal displacement of frame of submanifold, normalized 174 182 Infinitesimal displacement of frame of submanifold, tangentially degenerate 114 Infinitesimal displacement of frame of subspace 26 143 Infinitesimal displacement of frame of vector space 2 Infinitesimal displacement of tangential frame 20 Infinitesimal transformation of frame 1 Injective mapping 7 Inner geometry 206 Inscribed frame 214 215 Integrable distribution 144 151 Integrable distribution, asymptotic 146 Integrable distribution, conjugate 166 Integral, 1st 15 26 116 Integral, curve 73 144 145 166 280 283 Integral, curve of hyperdistribution 281 282 Integral, curve of manifold of hypercones 283 284 Integral, curve of Monge equation 280 281 Integral, element 12 105 139 153 164 Integral, element, one-dimensional 12 Integral, manifold 11—13 146 172 282 Integral, manifold, one-dimensional 11 Integral, submanifold 144 170 Intersection of algebraic images vi Intrinsic normalization of hyperband 262 Intrinsic normalization of hypersurface 222 224 Invariant 2nd normal 230 Invariant normal of hyperband 266 Invariant normalization of cone 207 Invariant normalization of hyperband 262 Invariant normalization of hypersurface 222 267 Invariant normalization of submanifold 199—205 207 Invariant normalization of surface 245 Invariant of surface 245 Invariant operation 18 Invariant plane 260 264 Invariant point 219 241 265 Invariant relative 4 159 160 194 199 207 224—226 230 233 242 246 263 265 Invariant subgroup 22 Invariant subspace 147 Invariant, 1st normal 230 Invariant, absolute 4 227 231 250 Invariant, dual frame 265 Invariant, form of general linear group 2 Invariant, form of group of centroprojective transformations 74 Invariant, form of stationary subgroup of element of normalized 175 Invariant, form of stationary subgroup of hyperplane 25 Invariant, form of stationary subgroup of point 24 183 Invariant, form of stationary subgroup of subspace 26 Invariant, frame 265 Invariant, hyperplane 265 Invariant, hyperquadric 23 219 274 Invariant, normalization 174 198 199 206 207 Inverse matrix 3 14 215 244 283 Inverse tensor 178 199 211 214 227 258 Invertible transformation 5 Involutive distribution 76 77 80 170 171 Irreducible conjugate, net 80 81 Isotropy transformation 22 Ivanova-Karatopraklieva 72 171 312 Ivlev 205 207 312 Izmailov 206 207 267 312 313 Jacobi matrix 7 33 Jensen vii 234 311 313 Jijtchenko 313 Juza 171 313 Kaehler metric v Kanitani v 266 267 313 314 Kernel 62—72 Khasin 71 314 Klein 314 Klingenberg 206 314 Kobayashi 296 307 314 Koenigs net 170 Koenigs net, axis of 171 Koenigs net, multidimensional vi 170 172 Konnov vii 212 300 Korovin 111 315 Kovantsov v 315 Koz'mina 111 315 Kronecker symbol 22 Kruglyakov v 70 171 315 324 Kubota 271 316 Kuiper 133 141 308 Lane v 209 316 Laplace network, closed 111 Laplace transform of, generalized conic submanifold 107 Laplace transform of, m-conjugate system 100 111 Laplace transform of, surface 90 91 111 Laplace transform of, surface, 2nd fundamental form of 98 Laptev v 207 209 224 266 267 316 317 Laugwitz 266 317 Law of transformation of covector 4 Law of transformation of tensor 4 115 Law of transformation of vector 3 Leave 275 legendre 295 Level hypersurface 7 Liber 206 207 317 Lichnerowicz 317 Lie 270 271 296 318 Lie, algebra 17 Lie, group 267 Lie, quadric 249 252 253 267 Liebmann 295 318 Line at infinity 173 Line, asymptotic 47 54 145 171 Line, asymptotic, rectilinear 155 158 159 Line, normal 173 Line, straight 155 Line, straight, double 31 Line, straight, projective 133 Linear element 207 Linear element, projective 214 234 251 266 267 Linear form of canonical tangent 251 Linear hull 138 Linear hypercomplex 282 Linear hypercomplex, special 282 Linear span 277 Linear subspace v Linear transformation 18 Little, J.A. 296 318 Little, J.B. vii 295 318 Local projective differential geometry vi Lopshits 206 318 Luchinin 207 312 Lumiste 146 171 205—207 267 318 319 m-conjugate system 78—80 82 83 98—101 104 106 m-conjugate system, 2nd Laplace transform of 101 m-conjugate system, conic 105 205 m-conjugate system, conic, existence of 105 m-conjugate system, conic, generalized 108 m-conjugate system, existence of 85 m-conjugate system, Laplace transform of 100 111 m-conjugate system, R 112 m-fold center 177 m-plane 40 Manifold of hypercones 256 283 Manifold of hypercones of 2nd order 284 Manifold of hypercones, asymptotic line of 284 Manifold of hypercones, basis forms of 283 Manifold of hypercones, con quadratic 284 Manifold of hypercones, equations of 283 Manifold of hypercones, integral curve of 283 284 Manifold of parameters 35 Manifold, algebraic 269 Manifold, analytic 5 Manifold, complex 5 Manifold, differentiable 5 19 33 34 Manifold, differentiable, class of 5 Manifold, differentiable, complex 5 Manifold, differentiable, connected 33 Manifold, differentiable, dimension of 5 Manifold, differentiable, structure equations of 14—15 Manifold, differentiable, tangent subspace to 5 Manifold, director 52 Manifold, integral 11—13 146 172 282 Manifold, integral, one-dimensional 11 Manifold, open simply connected 134 Manifold, smooth 269 Manifold, tangent space to 5 Mapping of tangent subspace 44 Mapping, analytic 7 Mapping, bijective 7 Mapping, class of 6 Mapping, differentiable 6 33 Mapping, differentiable, class of 6 Mapping, differentiable, Jacobi matrix of 7 Mapping, differentiable, nondegenerate 7 33 Mapping, Gauss 38 56 60 Mapping, geodesic 133 134 182 Mapping, Grassmann 27 Mapping, injective 7 Mapping, Meusner — Euler 40 Mapping, surjective 7 Mapping, Veronese 62—72 Matrix of quadratic form 215 Matrix, diagonal 82 121 122 Matrix, identity 82 Matrix, inverse 3 14 215 244 283 Matrix, Jacobi 7 33 Matrix, rank of 84 149 Matrix, square 82 Matrix, symmetric 113 138 Maurer — Cart an equations 10—11 Maximum rank d-web 279 Mean square 201 203 Method of exterior differential forms vii Method of moving frames vii Method tensorial vii Metric 56 Metric deformation 141 Metric degenerate 172 Metric property v 173 Metric space 267 Metric structure 141 Meusner — Euler mapping 40 Mihailescu v 319 Minimal normalization 176 178 Minor 28 Minor, diagonal 263 Mixed tensor 39 Module 8 Module, free 8 Module, generator of 8 Monge equation 271 279 280 Monge equation, asymptotic curve of 281 Monge equation, integral curve of 280 Monge equation, osculating plane of 280 Monge variety 279 280 296 Monge variety, degenerate 279—282 Monge variety, nondegenerate 280 Monosystem 171 Moving frame of 2-submanifold 287 Moving frame of conjugate net 73 103 Moving frame of curve 53 Moving frame of distribution of planes 143 Moving frame of Grassmannian 36 Moving frame of hyperband 256 Moving frame of hypercone 279 282 Moving frame of hyperdistribution 281 Moving frame of hypersurface 209 Moving frame of pair of hypersurfaces 272 Moving frame of Segre variety 37 Moving frame of submanifold 34 Moving frame of surface 90 91 245 252 Moving frame of Veronese surface 291 Moving frames method vii Multicomponent system of asymptotic distributions 150 171 Multicomponent system of asymptotic distributions, complete 150 Multidimensional generalization of hyperband 268 Multidimensional generalization of Koenigs net vi 170 172 Multidimensional generalization of Reiss theorem 271 Multidimensional projective differential geometry v vi Multidimensional web 276 296 Multiple component 120—123 Multiplication, exterior 8
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