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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
Product, exterior 18 36 Product, scalar 22 132 227 Product, symmetric tensor 37 Projection 278 Projection of Segre variety 150 Projective algebraic variety 278 Projective connection 205 207 266 267 Projective coordinates of point 28 29 35 Projective curvature tensor 180 Projective deformation 234 240 254 255 267 Projective differential geometry, local vi Projective differential geometry, multidimensional v vi Projective equivalence 234 240 Projective frame 18 Projective linear element 214 234 251 266 267 Projective normal 250 Projective property v Projective realization 134 Projective space 17 35 61 133 174 206 209 234 256 266 269 271 279 285 Projective space, 3-dimensional v 209 Projective space, dual 20 119 210 Projective space, infinitesimal displacement of frame of 19 20 287 Projective space, structure equations of 19 244 288 Projective straight line 133 Projective structure 141 Projective subspace 116 143 Projective subspace, equations of 143 Projective theory of surfaces 267 Projective transformation 18 240 244 254 Projectively, deformable surface 267 Projectively, equivalent hypersurfaces 234 240 254 Projectively, Euclidean connection 181 Projectivization 23 41 42 51 56 62 70 123 258 Projectivization of Segre cone 28 52 Projectivization of tangent subspace 35 36 Projectivization, center of 23 Prolongation, differential 161 Proper domain 134 Pseudoconformal geometry 256 Pseudoconformal space 256 Pseudoconformal structure 212 Pseudocongruence 129 130 174 175 183 188 190 222 Pseudocongruence, element of 129 Pseudocongruence, focal net of 130 Pseudocongruence, focal net of, holonomic 130 Pseudocongruence, focus cone of 130 Pseudocongruence, infinitesimal displacement of frame of 129 Pseudofocus 206 Pseudofocus, strong 190 Pyasetsky 206 322 Quadrangle 77 Quadrangle, closed 77 Quadratic form 272 277 Quadratic form, exterior 8 Quadratic hyperband 256 265 Quadratic hypercomplex 284 Quadratic normalization 177 178 180 185 Quadric(s) 158 159 162 251 Quadric(s), Darboux 249 Quadric(s), Fubini 249 Quadric(s), generating 172 Quadric(s), imaginary 22 Quadric(s), Lie 249 252 253 267 Quadric(s), pencil of 249 Quadric(s), Wilczynski — Bompiani 249 Quadrilateral 2-web 77 Quasi-Laplace transform 98 100 112 Quasi-Laplace transform of submanifold carrying conjugatenet 100 112 Quasi-Laplace transform of submanifold of codimension 2 101 Quasiasymptotic curve 71 111 171 Quasiasymptotic submanifold 71 111 Quasiconjugate directions 64 Quasitensor 115 127 129 183 224 225 227 230 233 259 Quasitensor, normal 229 230 Quasitensor, trace-free part of 183 Quotient 17 r-fold component 126 127 Rank of d-web 279 Rank of matrix 84 149 Rank of system of points 81 83 202 Rank of tangentially degenerate submanifold 96 166 Rank of tensor 295 Real asymptotic direction 67 Real conjugate net 159 Real domain vi 234 Real numbers field 1 Real part of focus variety 134 Real singularity 118 141 Realization, projective 134 Rectifiability condition 269 Rectilinear 3-web 275 Rectilinear 3-web, hexagonal 276 Rectilinear asymptotic line 155 158 159 Rectilinear congruence 129 Rectilinear generator 30 137 158 171 Rectilinearity 158 Reduced normal subspace 48 Reduced normal subspace, 1st 43 81 Reduced normal subspace, 2nd 45 81 Regular hyperband 256 268 Regular point 7 118 Regular submanifold 134 Regular submanifold, complete 137 Reiss 271 322 Reiss condition 279 Reiss theorem 270 271 278 295 Reiss theorem, 1st generalized 272 Reiss theorem, 2nd generalized 277 Reiss theorem, multidimensional generalization of 271 Reiss-type relations 295 Relative invariant 4 159 160 194 207 224—226 230 233 242 246 263 265 Relative nullity index 133 141 Relative nullity index, constant 133 134 Relative tensor 4 212 213 240 242 246 259 Relative vector 4 Relatively invariant form 212 214 233 250 Representation, faithful 74 Representation, space of GL(n) 16 Restriction of differential 6 16 147 Ricci identity 179 Ricci tensor 181 185 186 Ricci tensor, alternated 185 Ricci-flat connection 186 Riemannian space of constant curvature 72 132 134 141 206 Riemannian space of constant curvature, 3-dimensional 135 Riemannian space of constant curvature, elliptic 23 Riemannian space of constant curvature, Euclidean 21 22 134 135 141 173 205 221 259 Riemannian space of constant curvature, geodesic of 133 Riemannian space of constant curvature, hyperbolic 134 Riemannian submanifold 133 Rigidity 234 267 Rigidity problem 234 267 Rigidity theorem 234 Ring of differentiable functions 9 Ring of smooth functions 8 Rosenfeld 322 Roth 324 Ruled, hypersurface 171 Ruled, submanifold 94 Ruled, surface 29 156 158 171 235 251 Ruled, surface of 2nd order 30 Ruscior 71 171 322 Rybchenko 171 315 Ryzhkov 110 112 141 171 300 322 Sacksteder 135 141 323 Safaryan 141 282 295 323 Sasaki 212 266 296 323 Savelyev 141 323 Scalar function 110 Scalar product 22 132 227 Scheffers 270 271 276 295 318 323 Schouten 72 205 266 324 Secondary forms 6 242 Secondary parameters 6 147 260 263 283 294 Segre cone 28 Segre cone, dimension of 29 Segre cone, director manifold of 52 Segre cone, generator of 28 Segre cone, projectivization of 29 52 Segre theorem 56 59 122 127 259 Segre theorem, generalized 81—82 85 111 128 202 Segre type theorem 71 111 Segre variety 29 34 37 41 51 52 71—72 150 158 Segre variety, 2nd fundamental form of 41 158 Segre variety, 2nd osculating subspace of 41 Segre variety, dimension of 37 Segre variety, moving frame of 37 Segre variety, plane generator of 29 51 Segre variety, projection of 150 Segre variety, tangent subspace to 37 Segre, B. 111 295 323 Segre, C. 56 72 324 Self-conjugate 49 Self-dual 117 255 Semi-asymptotic net 61 111 Semi-conjugate net 71 111 Semple 324 Sequence of osculating subspaces 46 Several complex variables v Severi 72 324 Shadow submanifold 112 170 Shcherbakov v 171 324 Shelekhov 295 300 Shirokov, A.P. 324 Shirokov, P.A. 324 Shulikovskii v 324 Shveykin 206 324 325 Signature 54 Simple dyad 28 Simple pair of conjugate directions 68 Simple point 69 Simplex, autopolar 23 Simplex, coordinate 23 Simply connected manifold 134 Singular hyperplane 119 Singular point 7 36—38 118 132 Singularity 62 118 134 136 141 Singularity, real 118 141 Sisam 72 325 Skew-symmetric 27 51 52 Smirnov 102 112 325 Smooth manifold 269 Smooth plane curve 270 Smooth submanifold 285 Space of homeomorphisms 36 Space with affine connection 207 266 267 Space with degenerate metric 172 Space with projective connection 205 207 266 267 Space, affine 21 173 206 224 266 Space, affine, infinitesimal displacement of frame of 21 Space, affine, structure equations of 22 Space, affine, transformation of 21 Space, centroaffine 206 Space, conformed 21 23 256 267 Space, cotangent 19 Space, cotangent, basis of 6 19 Space, dual 131 Space, dual, infinitesimal displacement of frame of 131 Space, elliptic 134 135 Space, elliptic, 3-dimensional 135 Space, elliptic, absolute of 134 Space, equiaffine 206 Space, Euclidean v 21 22 134 135 141 173 205 221 259 Space, Euclidean, 3-dimensional 135 Space, Euclidean, 4-dimensional 135—136 Space, Euclidean, structure equations of 22 Space, Euclidean, transformation of 22 Space, homogeneous 267 Space, hyperbolic 134 135 Space, hyperbolic, 3-dimensional 135 Space, metric 267 Space, non-Euclidean v 5 21 23 177 Space, non-Euclidean, elliptic 23 134 135 Space, non-Euclidean, Euclidean v 21 22 134 135 141 173 205 221 258 Space, non-Euclidean, hyperbolic 134 135 Space, normal 173 Space, projective 17 35 61 133 174 206 209 234 256 266 269 271 279 285 Space, projective, 3-dimensional v 209 Space, projective, dual 20 119 210 Space, projective, infinitesimal displacement of frame of 19 20 287 Space, projective, structure equations of 19 244 288 Space, pseudoconformal 256 Space, Riemannian of constant curvature 72 132 134 141 206 Space, Riemannian of constant curvature, 3-dimensional 135 Space, Riemannian of constant curvature, geodesic of 133 Space, tangent 5 19 Space, tangent, basis of 19 Space, vector 1 8 17 131 214 227 Space, vector, basis of 1 35 Space, vector, infinitesimal displacement of frame of 2 Span, linear 277 Special linear group 18 Square matrix 82 Stationary subgroup of element of normalized 175 Stationary subgroup of hyperplane 25 Stationary subgroup of hyperplane, invariant form of 25 Stationary subgroup of planar element 35 Stationary subgroup of point 24 43 183 Stationary subgroup of point, invariant form of 24 183 Stationary subgroup of subspace 25 26 Stationary subgroup of subspace, invariant form of 26 Sternberg 325 Stipa 267 325 Stolyarov 205 268 325 326 Straight line(s) 155 Straight line(s), complex of 271 Straight line(s), congruence of 129 Straight line(s), double 31 Straight line(s), one-parameter family of 275 Straight line(s), parallel 21 Straight line(s), pencil of 137 275 Straight line(s), projective 133 Stratifiable pair, one-sided 187 188 206 Strong pseudofocus 190 Structure equations of affine space 22 Structure equations of differentiable manifold 14—15 Structure equations of Euclidean space 22 Structure equations of general linear group 10—12 Structure equations of projective space 19 244 288 Structure, conformal 212 Structure, metric 141 Structure, projective 141 Structure, pseudoconformal 212 Struik 72 323 SU v 266 268 326 327 Subbundle of 1st order moving frames of distribution 143 Subbundle of 4th order 237 242 Subbundle, base of 24 Subbundle, fiber of 24 Subbundle, parallel 205 Subgroup, invariant 22 Submanifold(s) v vi 33 92 174 Submanifold(s) of codimension 2 101 207 Submanifold(s) of codimension 2, conjugate net of 81 Submanifold(s) of codimension 2, invariant normalization of 207 Submanifold(s) of codimension 2, quasi-Laplace transform of 101 Submanifold(s) of codimension 3 101 Submanifold(s) of codimension 3, transformation of 101 Submanifold(s) of singular points of 118 Submanifold(s), 72
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