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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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Предметный указатель
Form(s), invariant of stationary subgroup of element of normalized 175 Form(s), invariant of stationary subgroup of hyperplane 25 Form(s), invariant of stationary subgroup of point 24 183 Form(s), invariant of stationary subgroup of subspace 26 Form(s), linear 250 Form(s), parallel vi 221 222 Form(s), Pfaffian 9 279 Form(s), principal 184 Form(s), quadratic 272 277 Form(s), quadratic exterior 8 Form(s), relatively invariant 212 214 233 250 Form(s), secondary 6 242 Form(s), trilinear 50 Formulas, derivational 2 FOURIER 295 Frame 1 Frame inscribed into cone 214 215 Frame of 1st order 34 Frame of 2nd order 42 Frame of 3rd order 238 241 Frame of 4th order 240—242 Frame, affine 21 22 Frame, bundle 5 16 Frame, bundle, dimension of 5 Frame, conjugate to cone 214 215 Frame, Finikov's 245 248—250 268 Frame, invariant 265 Frame, invariant of hyperband 265 Frame, invariant, dual 265 Frame, point 129 Frame, projective 18 Frame, subbundle of 3rd order 235 237 Frame, subbundle of 4th order 237 240 244 Frame, tangential 20 Frame, tangential infinitesimal displacement of 20 257 Frame, vectorial 18 Free module 8 Frobenius theorem 11 15 24 77 115 144 187 Fubini v 209 214 234 266 267 309 Fubini, net of surface 251 Fubini, projective linear element 214 234 251 266 267 Fubini, quadric 249 Function vii Function, analytic 7 Function, differentiable 33 Function, homogeneous 33 Function, homogeneous of 1st degree 34 Function, scalar 110 Fundamental form of Grassmannian 50 Fundamental form of image of Grassmannian 52 Fundamental object of 4th order 242 Fundamental object of 5th order 241 244 Fundamental tensor of order q 48 G-structure 295 G-structure, closed 295 Gardner 304 Gauss mapping 38 56 60 113 Gauss mapping of hypersurface 210 211 Gauss mapping, image of 38 56 59 113 Geidelman 112 309 General linear group vi 1 2 179 General linear group, invariant form of 2 6 General linear group, representation space of 16 General linear group, structure equations of 10—12 Generalized 1st Reiss theorem 272 Generalized 2nd Reiss theorem 277 Generalized conic submanifold 107—110 Generalized conic submanifold, construction of 108 109 Generalized conic submanifold, existence of 108 Generalized Segre theorem 81—82 85 111 128 202 Generating element 269 Generating quadric 172 Generator of cone 126 127 Generator of hypercone 131 279 280 Generator of hypersurface of rank one 116 Generator of module 8 Generator of Segre cone 28 Generator of submanifold 116—118 132 134 148 150 Generator, focus of 137 Generator, rectilinear 30 137 158 171 Geodesic 180 Geodesic of 133 Geodesic, equations of 180 Geodesic, mapping 133 134 182 Geodesic, torsion 205 Geodesic, transformation 180 Geometry, affine vi 221 Geometry, algebraic vi Geometry, conformal 256 Geometry, Euclidean v vi 221 Geometry, inner 206 Geometry, non-Euclidean v vi Geometry, projective differential v vi Geometry, pseudo conformal 256 Gluzdov 310 Godeaux 253 310 Godeaux surface 253 Goldberg 206 207 268 275 278 295 300 310 Goldsmith 304 Goldstein vii Gormasheva 112 302 Graf — Sauer theorem 276 295 Gramm, determinant 227 Gramm, matrix 202 Grassmann, 3-web 275 Grassmann, 3-web, 2-dimensional 276 Grassmann, 3-web, curvature tensor of 275 Grassmann, 3-web, GW(3,2,1) 275 276 Grassmann, 3-web, GW(3,2,r) 275 276 Grassmann, 3-web, hexagonal 275 276 Grassmann, algebra 8 Grassmann, coordinates 27 31 Grassmann, d-web 276 278 Grassmann, d-web 276 Grassmann, image 141 Grassmann, image, basis forms of 113 114 Grassmann, mapping 27 Grassmannian 26—28 34 35 38 50 70 210 272 275 277 280 Grassmannian basis form of 38 128 Grassmannian fundamental form of 50 Grassmannian image of 27 28 34 50 Grassmannian image of 2nd fundamental form of 51 Grassmannian image of 2nd osculating subspace of 51 Grassmannian image of asymptotic cone of 51 Grassmannian image of asymptotic cone of order it of 53 Grassmannian image of asymptotic direction of order k of 53 Grassmannian image of dimension of 35 36 Grassmannian image of fundamental form of order k of 52 Grassmannian image of osculating subspace of order k of 52 Grassmannian image of plane generator of 28 Grassmannian image of tangent subspace to 35 Grassmannian moving frame of 36 Grassmannian osculating subspace of 50 GrEen edge 248 Griffiths vi 31 70 234 267 278 295 296 304 308 310 311 Group of affine transformations 16 22 Group of centroprojective transformations 74 Group of conformal transformations 23 Group of motions 173 Group of projective transformations 18 19 Group of transformations of 1st order frames 35 Grove 205 311 Haantjes 205 264 323 Harmonic conjugate vector 267 Harmonic connection 176 185 186 Harmonic normalization 176—178 185 188 206 Harmonic pole 205 Harris vi 31 70 234 296 308 311 Havelka 267 311 Hexagonal rectilinear 3-web 276 Hexagonality condition 275 Hlavaty v 206 266 267 311 Hodge 311 Holonomic distribution 77 144 Holonomic net 76—77 123—125 Holonomic net, asymptotic 155 Holonomic net, conjugate 78 104 191 Holonomic net, focal 98 123—125 Holonomicity 71 80 111 125 Holst 295 312 Homogeneous algebraic polynomial 195 279 Homogeneous coordinates of hyperplane 20 Homogeneous coordinates of point 18 28 30 33 104 136 150 158 255 Homogeneous function 195 279 Homogeneous function of 1st degree 34 Homogeneous function, differentiable 33 Homogeneous parameters 29 34 Homogeneous space 267 Homomorphisms, space of 36 Horizontal distribution 16 183 Horizontal distribution, invariant 16 17 Horizontal form 25 Hull, linear 138 Hyperband vi 255 268 Hyperband, 1st normal of 262 Hyperband, 1st order neighborhood of 257 Hyperband, 2nd fundamental form of 256 Hyperband, 2nd normal of 262 Hyperband, 2nd order neighborhood of 257 259 260 Hyperband, 3rd order neighborhood of 262 264 Hyperband, 4th order neighborhood of 262 265 Hyperband, autodual 254 Hyperband, conic 259 260 262 263 266 Hyperband, Darboux tensor of 265 Hyperband, degenerate 268 Hyperband, dual-normalized 262 Hyperband, element of 255 260 Hyperband, infinitesimal displacement of frame of 257 Hyperband, intrinsic normalization of 262 Hyperband, invariant frame of 265 Hyperband, invariant normal of 266 Hyperband, invariant normalization of 262 Hyperband, moving frame of 256 Hyperband, multidimensional 268 Hyperband, nonquadratic 266 Hyperband, normalizing object of 264 266 Hyperband, normalizing plane of 260 261 Hyperband, one-dimensional 259 Hyperband, planar 259 262 263 266 Hyperband, quadratic 256 265 Hyperband, regular 256 268 Hyperband, support submanifold of 255 261 Hyperband, support submanifold of, plane generator of 255 257 260 261 Hyperband, tangential frame of 257 Hyperbolic point 61 Hyperbolic point of 2-dimensional submanifold 67 Hyperbolic space 134 135 Hyperbolic space, 3-dimensional 135 Hypercomplex 280 Hypercomplex, linear 282 Hypercomplex, linear, special 282 Hypercomplex, quadratic 284 Hypercone(s) 116 122 130 131 216 279 280 283 285 Hypercone(s), asymptotic of hyperband 279—284 Hypercone(s), cubic 215 Hypercone(s), distribution of 279 Hypercone(s), generator of 131 279 280 Hypercone(s), infinitesimal displacement of frame of 282 Hypercone(s), manifold of 256 283 Hypercone(s), manifold of 2nd order 284 Hypercone(s), manifold of asymptotic line of 284 Hypercone(s), manifold of basis forms of 282 Hypercone(s), manifold of conquadratic 284 Hypercone(s), manifold of equations of 283 Hypercone(s), manifold of integral curve of 283 284 Hypercone(s), moving frame of 131 279 282 Hypercone(s), nondegenerate 282 296 Hypercone(s), nondegenerate of 2nd order 296 Hypercone(s), tangent hyperplane to 284 Hypercone(s), vertex of 122 131 Hypercubic 31 275 Hyperdistribution 281 Hyperdistribution, algebraizability condition of 282 Hyperdistribution, asymptotic, hypercone of 281 282 Hyperdistribution, asymptotic, line of 281 282 Hyperdistribution, integral curve of 281 282 Hyperdistribution, moving frame of 281 Hyperplanar element 279 281 296 Hyperplane(s) 19 54 218 282 Hyperplane(s) at infinity 21 134 177 224 Hyperplane(s), algebraic family of 119 Hyperplane(s), basis 131 Hyperplane(s), double 131 Hyperplane(s), focus 119 130 Hyperplane(s), ideal 21 Hyperplane(s), improper 134 136 Hyperplane(s), invariant 265 Hyperplane(s), pencil of 282 Hyperplane(s), polar 261 Hyperplane(s), singular 119 Hyperplane(s), stationary subgroup of 25 Hyperplane(s), stationary subgroup of, invariant form of 25 Hyperplane(s), tangent 49 136 284 Hyperplane(s), tangent to hypercone 284 Hyperplane(s), tangent to hyperquadric 219 Hyperplane(s), tangent to hypersurface 211 216 272 Hyperquadric 23 112 130 177 216—222 234 256 269 274 275 Hyperquadric, 2nd normal of 177 Hyperquadric, convex 134 Hyperquadric, Darboux osculating 217 218 220 266 267 Hyperquadric, degenerate 130 132 Hyperquadric, imaginary 134 Hyperquadric, invariant 23 219 274 Hyperquadric, nondegenerate 23 Hyperquadric, Pluecker 28 Hyperquadric, positive definite 23 Hyperquadric, subnormal of 177 Hyperquadric, tangent hyperplane to 219 Hypersurface(s) 54 101 122 127 131 136 140 205 209 234 266 291 Hypersurface(s) in 135—136 Hypersurface(s) in 136 Hypersurface(s) in 290 Hypersurface(s) in 290 Hypersurface(s) in 291 Hypersurface(s) of rank one 116 Hypersurface(s) of rank one, generator of 116 Hypersurface(s) of second order 216 Hypersurface(s), 1st normal of 212 222 241 242 248 Hypersurface(s), 1st normal space of 53 Hypersurface(s), 1st reduced normal space of 53 Hypersurface(s), 2nd fundamental form of 53 210 212 221 233 244 272—275 277 278 291 Hypersurface(s), 2nd fundamental tensor of 54 218 221 Hypersurface(s), 2nd fundamental tensor of, covariant derivative of 221 Hypersurface(s), 2nd fundamental tensor of, indefinite 226 Hypersurface(s), 2nd fundamental tensor of, positive definite 226 242 Hypersurface(s), 2nd fundamental tensor of, rank of 54 Hypersurface(s), 2nd neighborhood of 233 Hypersurface(s), 2nd normal of 212 222 224 241 242 248 Hypersurface(s), 2nd order asymptotic direction on 230 Hypersurface(s), 3rd neighborhood of 211 225 227 233 Hypersurface(s), 4th neighborhood of 212 213 224 233 241 242 Hypersurface(s), 5th neighborhood of 232 233 241 242 Hypersurface(s), affine connection on 221 Hypersurface(s), algebraic 122 272 278 Hypersurface(s), asymptotic cone of 211 Hypersurface(s), asymptotic cone of 2nd order of 53 210 Hypersurface(s), asymptotic curve of 54 Hypersurface(s), asymptotic direction of 230 Hypersurface(s), basis form of 210 244 Hypersurface(s), canal 172
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