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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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Bundle, normal 178 182 183 Bundle, normal, affine connection in 183 Bundle, normal, base of 182 Bundle, tangent 5 178 183 Bundle, tangent, affine connection in 179 183 193 Bundle, tangent, element of 5 Bushmanova 230 304 Canal hypersurface 172 Canonical covector 254 Canonical form of fundamental form 76 Canonical pencil of 1st normals 250 267 Canonical pencil of 1st normals, classical 231 250 Canonical tangent 250 251 Canonical tensor, 1st 230 Cartan v—vii 71 83 89 111 141 151 171 205 234 240 267 304 305 Cartan, lemma 9 Cartan, line 250 Cartan, number 13 86 89 139 140 153 Cartan, test 13 14 86 89 105 139 141 191 Cartan, variety 83—84 111 207 Cartan, variety, invariant normalization of 207 Cartesian coordinate system 270 Cartesian coordinates 269 Casanova 305 Cauchy 295 Cech v 209 305 309 Cech, axis 248 Cenkl 266 305 Center of bundle of hyperplanes 121 122 189 Center of projectivization 23 Center, m-fold 177 Central normalization 177 179 185 205 Centroaffine space 206 Centroprojective connection 183 Centroprojective transformation 74 Chakmazyan 205 206 268 300 305—307 310 319 CHARACTER 13 86 139 140 153 Characteristic, direction 217 Characteristic, equation 120 188 Characteristic, form 139 Characteristic, plane 120 124 128 Characteristic, subspace 49 52 53 120 Chebyshev covector 225 Chebysheva 172 320 Chen 221 205 307 Chern v 111 133 141 278 295 296 304 307 308 Class of differentiable manifold 5 Class of differentiable mapping 6 Classical canonical pencil of 1st normals 230 250 Classical pencil of Darboux quadrics 249 Classification of conjugate nets 81—85 Classification of points 57 61 63 70 72 Classification of points of 64—67 Classification of points of 67—70 72 Closed form 10 Closed G-structure 295 Closed Laplace network 111 Closed quadrangle 77 Closed system 109 Cobasis 5 34 272 Cofactor 224 Coframe 20 Collinear vectors 17 18 290 Common Lie quadrics 252 253 Complementary subspace 44 277 Complete 2-component asymptotic distribution 148 Complete 2-component system 148 Complete 2-component system, doubly foliated 149 150 Complete 2-component system, integrable conjugate 169 Complete multicomponent, asymptotic distribution 150 171 Complete multicomponent, system 150 Complete object 244 Complete parabolic submanifold 134 141 Complete parabolic submanifold, regular 135 Complete parabolic submanifold, without singularities vi 134 Complete regular submanifold 137 Complete smooth curve 137 Completely integrable system 11 Complex domain vi 234 Complex manifold 5 Complex numbers, field of 1 Complex of straight lines 271 Complexes, T-pair of 112 Component, 126 127 Component, multiple 120—123 Cone(s) 104 119—122 126—128 135 168 170 207 214 Cone(s) of 2nd order 215 Cone(s), algebraic 130 Cone(s), asymptotic 47 Cone(s), asymptotic of 2nd order of hypersurface 210 Cone(s), asymptotic of image of Grassmannian 51 53 Cone(s), asymptotic of order q 48 Cone(s), asymptotic, filtration of 49 53 Cone(s), cubic 49 58 59 214 218 Cone(s), Darboux 214 218 267 Cone(s), distribution of 271 Cone(s), filtration of 49 53 Cone(s), focus 119 121 130 175 188 189 261 Cone(s), focus, equation of 122 Cone(s), focus, vertex of 121 261 Cone(s), generator of 126 127 Cone(s), invariant normalization of 207 Cone(s), Segre 28 Cone(s), Segre, dimension of 29 Cone(s), Segre, director manifold of 52 Cone(s), Segre, generator of 28 Cone(s), Segre, projectivization of 29 52 Cone(s), vertex of 116 119 120 126—128 135 168 170 Conformal geometry 256 Conformal space 21 23 256 267 Conformal structure 212 Congruence of planes 90 128 129 174 175 181 183 194 222 Congruence of planes of straight lines 129 Congruence of planes, conjugate to 189 190 Congruence of planes, focal net of 129 Congruence of planes, holonomic 129 Congruence of planes, infinitesimal displacement of frame of 128 Congruence of planes, moving frame of 128 Congruence of planes, rectilinear 129 Conic(s) 56 57 Conic(s), conjugate net vi 199 Conic(s), double 31 Conic(s), hyperband 259 260 262 263 266 Conic(s), m-conjugate system 105 205 Conic(s), m-conjugate system, existence of 105 Conic(s), net 112 201 Conic(s), pencil of 57 Conic(s), submanifold 104 112 168 Conic(s), submanifold, 2-dimensional 109 110 Conic(s), submanifold, equations of 104 110 Conic(s), submanifold, generalized 107—110 Conic(s), submanifold, vertex of 168 Conic(s), surface 108 Conic(s), system 107 Conisecant plane 30 31 Conjugate, directions 94 Conjugate, directions of 123 Conjugate, directions of 2nd order 49 57 63 Conjugate, directions of 3-submanifold 58 67—70 Conjugate, directions of 3rd order 50 71 87 111 Conjugate, directions of hypersurface 54 211 Conjugate, distributions 93 94 98 Conjugate, distributions, complementary 94 Conjugate, distributions, integrable 166 Conjugate, foliations 94 Conjugate, frame 214 215 Conjugate, net 55 58 73 74 77 83 85 100 103 159 191 Conjugate, net of 2-submanifold 60 Conjugate, net of 3-submanifold 57 Conjugate, net of order k 76 Conjugate, net on surface 55 91 92 Conjugate, net, 2nd fundamental form of 74 Conjugate, net, 2nd osculating subspace of 75 Conjugate, net, conic vi 199 Conjugate, net, holonomic 78 104 191 Conjugate, net, infinitesimal displacement of frame of 103 Conjugate, net, irreducible 80 81 Conjugate, net, moving frame of 73 104 Conjugate, net, nonholonomic 80—81 Conjugate, net, real 159 Conjugate, normalization 189 206 Conjugate, subspace 215 Conjugate, system 78—80 Conjugate, system of distributions 165 Conjugate, system, 2-component 112 165 171 Conjugate, system, conic 205 Conjugate, system, conic 2-component 169 Conjugate, system, multicomponent 171 Conjugate, vectors 214 Conjugate-harmonic normalization 191 206 Conjugate-harmonic normalization, existence of 191 Connection, affine 16 17 184 Connection, base of 178 Connection, centroprojective 183 Connection, equiaffine 186 Connection, flat normal 187—190 206 Connection, form 17 179 Connection, harmonic 176 186 Connection, normal 183 186 205 206 Connection, normal, affine 184 Connection, normal, curvature form of 184 185 205 Connection, normal, flat 187—190 206 Connection, projective 205 207 266 267 Connection, projectively Euclidean 181 Connection, Ricci-flat 186 Connection, torsion and curvature tensors of 17 179 Connection, torsion-free projective 205 Connection, typical fiber of 178 Conquadratic 285 Convex hyperquadric 134 Coordinate(s) of tangent vector 14 48 Coordinate(s), cartesian 269 Coordinate(s), differentiable 33 Coordinate(s), Grassmann 27 31 Coordinate(s), homogeneous of hyperplane 20 Coordinate(s), homogeneous of point 18 28 30 33 104 136 150 158 255 Coordinate(s), nonhomogeneous 255 Coordinate(s), Pluecker 28 Coordinate(s), projective of point 28 29 35 Coordinate(s), simplex 36 Coordinate(s), system, affine 278 Coordinate(s), system, Cartesian 270 Coordinate(s), tangential 20 49 261 Coplanar 72 Correlative transformation 116 117 123 255 275 276 Correspondence, degenerate 71 Cotangent, bundle 6 Cotangent, space 19 Cotangent, space, basis of 19 Covariant derivative 184 Covariant derivative of 2nd fundamental tensor 221 Covariant differential of 2nd fundamental tensor 221 Covariant differentiation 184 Covector 49 Covector, canonical 254 Covector, Chebyshev 225 Covector, differential equations of 3 Covector, field 8 Covector, law of transformation of 4 Cubic, cone of directions 49 58 59 214 218 Cubic, curve 58 59 Cubic, form 213 Cubic, hyper surface 286 Cubic, hypercone 215 Cubic, plane curve 70 Cubic, symmetroid 31 67—69 286 287 Curvature tensor 17 179 Curve 53 104 105 137 253 Curve of 2nd order 31 269 Curve of 3rd class 276 Curve of 3rd order 276 Curve of 4th order 31 Curve, algebraic 270 Curve, asymptotic 47 54 145 171 Curve, asymptotic of 2nd order 48 Curve, asymptotic of hyperdistribution 281 282 Curve, asymptotic of hypersurface 211 Curve, asymptotic of manifold of hypercones 284 Curve, asymptotic of Monge equation 281 Curve, asymptotic of order q 48 71 Curve, asymptotic of surface 251 Curve, asymptotic, osculating 2-plane of 47 Curve, asymptotic, rectilinear 155 158 159 Curve, complete smooth 137 Curve, cubic 58 59 Curve, cubic, plane 70 Curve, infinitesimal displacement of frame of 53 Curve, integral 73 144 145 166 280 283 Curve, integral of hyperdistribution 282 283 Curve, integral of manifold of hypercones 283 284 Curve, integral of Monge equation 280 281 Curve, moving frame of 53 Curve, nonplanar 116 Curve, osculating 2-plane of 41 49 180 203 Curve, osculating plane of order q of 48 Curve, osculating subspace of 53 Curve, plane 269 Curve, plane of 2nd order 64 Curve, plane, smooth 270 Curve, quasiasymptotic 71 111 171 Curve, tangent subspace to 53 Curvilinear 2-web 77 Cylinder 135 d-web 276 278 279 d-web of maximum rank 279 d-web, Grassmann 278 d-web, Grassmann, algebraic 276 d-web, rank of 279 Darboux 111 308 Darboux, cone 214 218 267 Darboux, direction 267 Darboux, invariant 111 Darboux, line of surface 251 Darboux, osculating hyperquadric 218 220 232 266 267 Darboux, osculating hyperquadric, pencil of 218 223 231 235 237 249 267 Darboux, quadric 249 Darboux, quadric, classical pencil of 249 Darboux, tensor 112 171 213 218 220 222 225—228 244 266 269 Darboux, tensor of hyperband 265 Darboux, tensor, degenerate 227 Darboux, tensor, nondegenerate 226 Deformation, metric 141 Deformation, projective 234 240 254 255 267 Degen 112 171 172 308 Degeneracy of surface 251 Degenerate, bisecant variety 285 286 Degenerate, correspondence 71 Degenerate, focus variety 118 Degenerate, hyperband 268 Degenerate, hyperquadric 130 132 Degenerate, metric 172 Degenerate, Monge variety 279—281 Derivational formulas 2 Derivative, covariant 184 Derivative, covariant of 2nd fundamental tensor 221 Determinant of tensor 236 Determinant, submanifold 29 30 67 130 131 Determinant, submanifold, algebraic 53 Deviation of osculating 2-plane 203 Diagonal form 75 243
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