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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
Diagonal matrix 82 121 122 Diagonal minor 263 Diagonalizable operator 96 Dieudonne 31 308 Differentiable coordinates 33 Differentiable function 33 Differentiable function, homogeneous 33 Differentiable manifold 5 19 33 34 Differentiable manifold, algebraic 269 Differentiable manifold, class of 5 Differentiable manifold, complex 5 Differentiable manifold, connected 33 Differentiable manifold, dimension of 5 Differentiable manifold, structure equations of 14—15 Differentiable manifold, tangent subspace to 5 Differentiable mapping 6 33 Differentiable mapping, class of 6 Differentiable mapping, Jacobi matrix of 7 Differentiable mapping, nondegenerate 7 33 Differential form vii Differential form, exterior vii Differential, 1-form 8 Differential, covariant 221 Differential, equations of covector 3 Differential, equations of subspace 143 Differential, equations of tensor 3 Differential, equations of vector 3 Differential, exterior 10 144 179 211 257 Differential, invariant, absolute 214 233 Differential, neighborhood of, 1st order 156 257 Differential, neighborhood of, 2nd order 39 48 97 98 233 259 260 Differential, neighborhood of, 3rd order 44 92 198 211 213 225 227 233 262—264 Differential, neighborhood of, 4th order 198 212 213 224 233 242 262 265 Differential, neighborhood of, 5th order 232 233 241 242 Differential, neighborhood of, order r 207 Differential, operator 4 184 200 211 258 Differential, p-form 207 Differential, prolongation 161 Differential, restriction of 6 16 47 Differential, total 21 105 176 186 Differentiation, covariant 184 Differentiation, exterior 10 Differentiation, exterior, invariance of 10 Differentiation, relative to secondary parameters 6 24 26 147 260 263 283 Dimension of 2nd normal subspace 43 Dimension of 2nd osculating subspace 75 Dimension of bundle of 2nd fundamental forms 40 Dimension of bundle of 3rd fundamental forms 45 Dimension of differentiable manifold 5 Dimension of frame bundle 5 Dimension of image of Grassmannian 35 36 Dimension of Segre cone 29 Dimension of Segre variety 37 Direct product 30 Direction(s), asymptotic of 2nd order 48 57—71 92 94 145 230 Direction(s), asymptotic of image of Grassmannian 53 Direction(s), asymptotic of order q 48 71 Direction(s), asymptotic, double 66 70 Direction(s), asymptotic, real 67 Direction(s), asymptotic, threefold 69 Direction(s), characteristic 217 Direction(s), conjugate 94 Direction(s), conjugate of 123 Direction(s), conjugate of 2nd order 49 57 63 Direction(s), conjugate of 3-submanifold 58 67—70 Direction(s), conjugate of 3rd order 50 71 87 111 Direction(s), conjugate of hypersurface 54 211 Direction(s), conjugate, double pair of 68 Direction(s), conjugate, simple pair of 68 Direction(s), conjugate, threefold pair of 66 Direction(s), Darboux 267 Direction(s), double asymptotic 66 70 Direction(s), focal 98 123 129 130 137 188 Direction(s), mutually conjugate 55 59 60 73 Direction(s), quasiconjugate 64 Director manifold 52 Discriminant 2nd order of hypersurface 210 224 246 Discriminant of hypersurface 226 Distribution(s) 143 147 183 Distribution(s) of cones 271 Distribution(s) of hypercones 279 Distribution(s) of hypercones of 2nd order 295 Distribution(s) of hyperplanar elements 279 281 296 Distribution(s) of linear elements 207 Distribution(s) of planes 143 Distribution(s) of planes, moving frame of 143 Distribution(s), 2-dimensional 77 Distribution(s), asymptotic 145 171 Distribution(s), asymptotic, complete 2-component 148 Distribution(s), asymptotic, complete multicomponent 150 171 Distribution(s), asymptotic, integrable 146 Distribution(s), asymptotic, one-dimensional 151 Distribution(s), conjugate 93 94 98 Distribution(s), conjugate, complementary 94 Distribution(s), holonomic 77 144 Distribution(s), horizontal 16 183 Distribution(s), horizontal, invariant 16—17 Distribution(s), integrable 144 151 166 Distribution(s), involutive 76 77 80 170 171 Distribution(s), one-parameter 93 do Carmo 296 307 Domain, complex vi 234 Domain, proper 134 Domain, real vi 234 Double, asymptotic direction 66 70 Double, conic 31 Double, focus 137 Double, foliation 149 Double, hyperplane 131 Double, pair of conjugate directions 68 Double, point 69 Double, straight line 31 Double, symmetric transformation 62 Double, tangency 69 Double, tangent plane 249 Doubly foliated submanifold 83 Dual basis 5 Dual focal image 119 Dual frame and coframe 20 265 Dual image of hypersurface 210 Dual image of hypersurface, 2nd fundamental form of 211 Dual image of hypersurface, generating element of 210 Dual normalization 268 Dual normalization of hyperband 264 Dual normalizing planes 260 Dual projective space 20 119 210 Dual space 131 Dual space, infinitesimal displacement of frame of 131 Dual-normalized hyperband 262 Duality 129 Duality principle 260 262 Dyad, simple 28 Edge of regression 100 Edge of regression of 118 Egorov, transformation vi Eigenvalue 120 188 243 244 Eisenbud 308 Element of hyperband 255 260 Element of normalized 175 Element of normalized 179 Element of pseudocongruence 129 Element of tangent bundle 5 Element of volume 186 Element, generating 269 Element, hyperplanar 279 281 295 Element, integral 12 105 139 153 164 Element, integral, one-dimensional 12 Element, linear projective 207 Element, natural normalizing 173 Element, planar 35 Element, planar, stationary subgroup of 35 Elliptic point 61 Elliptic point of 2-dimensional submanifold 67 Elliptic space 134 135 Elliptic space, 3-dimensional 135 Elliptic space, absolute of 134 Elliptic transformation 23 Embedding 30 Embedding, symmetric 30 Embedding, Veronese 296 Engel 271 295 308 Envelope 60 117 119 122 Equation(s) of conic submanifold 104 110 Equation(s) of focus cone 122 Equation(s) of focus variety 122 Equation(s) of geodesic 180 Equation(s) of manifold of hypercones 283 Equation(s), algebraic vi 94 95 99 204 Equation(s), bilinear 223 Equation(s), characteristic 120 188 Equation(s), differential of covector 3 Equation(s), differential of subspace 143 Equation(s), differential of tensor 3 Equation(s), differential of vector 3 Equation(s), integral curve of 280 Equation(s), Maurer — Cartan 10—11 Equation(s), Monge 271 279 280 Equation(s), Monge, asymptotic curve of 281 Equation(s), osculating plane of 280 Equation(s), Pfaffian 281 Equation(s), Pfaffian, system of 11 12 Equation(s), structure of affine space 22 Equation(s), structure of differentiable manifold 14—15 Equation(s), structure of Euclidean space 22 Equation(s), structure of general linear group 10—12 Equation(s), structure of projective space 19 244 288 Equiaffine connection 186 Equiaffine space 206 Equipped submanifold 174 Equivalence classes 23 Equivalence projective 234 240 Equivalence relation 17 23 35 Ermakov 207 308 309 Euclidean geometry v vi 221 Euclidean space v 21 22 134 135 141 173 205 221 259 Euclidean space, 3-dimensional 135 Euclidean space, 4-dimensional 135—136 Euclidean space, structure equations of 22 Euclidean space, transformation of 22 Euler theorem 195 Exact form 10 Existence of conic m-conjugate system 105 Existence of conjugate harmonic normalization 191 Existence of generalized conic m-conjugate system 108 Existence of m-conjugate system 85 Existence of Peterson submanifold 108 Extension, natural 134 Exterior, differential 10 144 179 211 257 Exterior, differential, forms method vii Exterior, differentiation 10 Exterior, differentiation, invariance of 10 Exterior, multiplication 8 Exterior, p-form 8 9 Exterior, p-form, value of 9 Exterior, product 18 36 Exterior, quadratic form 8 Face 36 Faithful representation 74 Family of hypercones 131 Family of plane generators 117 149 Family of planes 147 207 Family, focal 95 96 101 112 Fernandez 266 309 Fiber form 15 25 Fiber form of normal bundle 182 Fiber of subbundle 24 Fibration 291 Field of complex numbers 1 Field of real numbers 1 Field, tensor vii 6 Field, vector 6 Filtration of asymptotic cones 49 53 Filtration of submanifolds 62 67 Finikov v 209 245 309 Finikov's frame 245 248 249 250 268 Flat normal connection 187—190 206 Focal direction 98 123 129 130 137 183 Focal family 95 96 101 112 Focal image 121—123 129 Focal image, dual 119 Focal net 97 98 101 123 124 Focal net of 118 123 Focal net of congruence 129 Focal net of pseudocongruence 130 Focal net, holonomic 98 123—125 Focal net, nonholonomic 123 124 128 Focal point 117 Focus 90 93—96 99 104 117 129 132 204 Focus of generator 137 Focus variety 117 120 121 123 126 129 132 134 166 168 175 177 184 194 261 Focus variety of 1st normal of 175 188 189 Focus variety of generator 166 Focus variety, degenerate 118 Focus variety, equation of 122 Focus variety, imaginary 137 Focus variety, real part of 134 Focus, cone 119 121 175 188 189 261 Focus, cone, equation of 122 Focus, cone, vertex of 121 261 Focus, double 137 Focus, hyperplane 119 130 Foliation(s) 7 115 Foliation(s), conjugate 94 Foliation(s), double 149 Foliation(s), one-dimensional 92 94 Form(s), 2nd fundamental of 2-submanifold 59—61 64—66 289 Form(s), 2nd fundamental of conjugate net 73 Form(s), 2nd fundamental of hyperband 258 Form(s), 2nd fundamental of hypersurface 53 210 212 244 272—275 277 Form(s), 2nd fundamental of image of Grassmannian 51 Form(s), 2nd fundamental of Laplace transform 98 Form(s), 2nd fundamental of Segre variety 41 158 Form(s), 2nd fundamental of submanifold 40 42 47 71 165 169 277 Form(s), 2nd fundamental of surface 91 92 288 Form(s), 2nd fundamental of Veronese variety 41 287 Form(s), 2nd fundamental, parallel vi 221 222 Form(s), 2nd fundamental, parallelism of 221 Form(s), 2nd fundamental, pencil of 57 Form(s), 3rd fundamental 44 49 153 Form(s), 3rd fundamental of submanifold with conjugate net 75 Form(s), base 14 35 Form(s), basis 35 Form(s), basis of Grassmannian 38 128 Form(s), basis of hypersurface 210 244 Form(s), bilinear 49 132 Form(s), canonical 76 Form(s), characteristic 139 Form(s), closed 10 Form(s), connection 17 179 Form(s), cubic 213 Form(s), diagonal 75 243 Form(s), differential vii Form(s), differential exterior vii Form(s), exact 10 Form(s), exterior quadratic 8 Form(s), fiber 15 25 Form(s), fiber of normal bundle 182 Form(s), horizontal 25 Form(s), invariant of general linear group 2 Form(s), invariant of group of centroprojective transformations 74
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