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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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Multipoint condition 270 Muracchini v 71 72 111 171 319 320 Musso 234 313 Mutually conjugate directions 55 59 60 73 Mutually inverse matrices 3 14 215 242 283 Mutually inverse tensors 202 214 258 Natural basis 14 Natural extension 134 Natural normalizing element 173 Neighborhood (differential) of 1st order 156 257 Neighborhood (differential) of 2nd order 39 48 97 98 233 251 259 260 Neighborhood (differential) of 3rd order 44 92 198 211 213 225 227 233 262 264 Neighborhood (differential) of 4th order 198 212 213 224 233 242 262 265 Neighborhood (differential) of 5th order 232 233 241 242 Neighborhood (differential) of order r 207 Net(s) 73 76 Net(s) of torses 123 Net(s), asymptotic holonomic 155 Net(s), classification of 81—85 Net(s), conic 112 201 Net(s), conjugate 55 58 73 74 77 83 85 101 103 159 191 Net(s), conjugate of 2-submanifold 60 Net(s), conjugate of 3-submanifold 57 Net(s), conjugate of order k 76 Net(s), conjugate on surface 55 91 92 Net(s), conjugate, 2nd fundamental form of 74 Net(s), conjugate, 2nd osculating subspace of 75 Net(s), conjugate, conic vi 199 Net(s), conjugate, holonomic 78 104 191 Net(s), conjugate, infinitesimal displacement of frame of 103 Net(s), conjugate, irreducible 80—81 Net(s), conjugate, moving frame of 73 103 Net(s), conjugate, nonholonomic 80 81 Net(s), conjugate, real 159 Net(s), focal 97 98 101 123 124 Net(s), focal of 118 123 Net(s), focal of congruence 129 Net(s), focal of pseudo congruence 130 Net(s), focal, holonomic 98 123—125 Net(s), focal, nonholonomic 123 124 128 Net(s), Fubini 251 Net(s), holonomic 76 77 Net(s), Koenigs 170 Net(s), Koenigs, multidimensional generalization of vi 170 Net(s), semi-asymptotic 61 111 Net(s), semi-conjugate 71 111 Nomizu 296 314 320 Non-Euclidean geometry v Non-Euclidean space v 21 23 177 Non-Euclidean space, elliptic 23 134 135 Non-Euclidean space, Euclidean v 21 22 134 135 141 173 205 221 259 Non-Euclidean space, hyperbolic 134 135 Nondegenerate differentiable mapping 7 Nondegenerate hypercone 282 296 Nondegenerate hypercone of 2nd order 296 Nondegenerate hyperquadric 23 Nondegenerate Monge variety 280 Nondegenerate tensor 257 258 Nondegenerate tensor, Darboux 226 Nondegenerate tensor, symmetric 177 Nonholonomic net, conjugate 80 81 Nonholonomic net, focal 123 124 128 Nonhomogeneous coordinates 255 Nonhomogeneous parameters 34 Nonplanar curve 116 Nonquadratic hyperband 266 Nonsingular point 33 34 Nonsingular tensor 197 Norden v 172 205 206 225 230 304 320 Normal(s), 1st 174 250 Normal(s), 1st invariant 230 Normal(s), 1st of hyperband 262 Normal(s), 1st of hypersurface 212 222 239 242 248 Normal(s), 1st subspace 36 40 53 61 147 167 Normal(s), 1st, pencil of 231 250 Normal(s), 1st, subnormal of 174 175 193 Normal(s), 2nd invariant 230 Normal(s), 2nd, pencil of 231 Normal(s), affine vi 224 268 Normal(s), bundle 178 182 183 Normal(s), bundle, affine connection in 183 Normal(s), bundle, base of 182 Normal(s), connection vi 183 186 205 206 Normal(s), connection, affine 184 Normal(s), connection, curvature form of 184 185 187 205 Normal(s), connection, flat 187—190 206 Normal(s), line 173 Normal(s), projective 250 Normal(s), quasitensor 228 Normal(s), space 173 Normal(s), subspace, 1st 36 40 54 55 61 84 147 167 Normal(s), subspace, 1st reduced 42 54 55 81 167 206 Normal(s), subspace, 2nd 42 45 Normal(s), subspace, 2nd reduced 45 81 Normal(s), subspace, reduced 48 Normal(s), Voss 111 Normalization(s), affine 177 180 185 Normalization(s), axial 205 Normalization(s), central 177 179 185 205 Normalization(s), condition 18 20 25 236 Normalization(s), conjugate 189 206 Normalization(s), conjugate-harmonic 191 206 Normalization(s), conjugate-harmonic, existence of 191 Normalization(s), dual 267 Normalization(s), dual of hyperband 264 Normalization(s), harmonic 176—178 185 188 206 Normalization(s), intrinsic 192 193 Normalization(s), intrinsic of hyperband 262 Normalization(s), intrinsic of hypersurface 222 224 Normalization(s), invariant 174 198 199 206 207 Normalization(s), invariant of cone 207 Normalization(s), invariant of hyperband 262 Normalization(s), invariant of hypersurface 222 267 Normalization(s), invariant of submanifold 199—205 207 Normalization(s), minimal 176—178 Normalization(s), pencil of 231 Normalization(s), quadratic 177 178 180 185 Normalized submanifold 174 206 266 Normalized submanifold of codimension 2 205 206 Normalized submanifold, element of 174 Normalized submanifold, infinitesimal displacement of frame of 174 282 Normalized submanifold, tangentially degenerate 205 Normalizing element, natural 173 Normalizing, object of 1st kind 193 197 198 222 241 Normalizing, object of 2nd kind 193 197 198 200 222 241 Normalizing, object of hyperband 264 266 Normalizing, object of hypersurface 224 227 Normalizing, plane 260 261 Normalizing, plane of hyperband 260 261 Normalizing, plane, dual 260 Object, complete 244 Object, fundamental of 4th order 242 Object, fundamental of 5th order 241 244 Object, normalizing of 1st kind 193 197 198 222 241 Object, normalizing of 2nd kind 193 197 198 200 222 241 Object, normalizing of hyperband 264 266 Object, normalizing of hypersurface 224 227 One-dimensional, asymptotic distribution 151 One-dimensional, foliation 92 94 One-dimensional, hyperband 259 One-dimensional, integral element 12 One-dimensional, integral manifold 11 One-parameter, distribution 93 One-parameter, family of 2nd degree curves 30 One-parameter, family of generators 123 One-parameter, family of pairs of conjugate directions 66 One-parameter, family of pencils of straight lines 275 One-parameter, family of planes 60 122 147 One-parameter, family of straight lines 100 275 One-point condition 269 One-sided stratifiable pair 187 188 206 Operation, invariant 18 Operator of covariant differentiation 184 Operator, 4 184 200 211 258 Operator, diagonalizable 96 Operator, van der Waerden — Bortolotti 184 Orbit 74 Osculating, 2-plane of asymptotic curve 47 Osculating, 2-plane of curve 41 49 180 203 Osculating, hyperquadric 217 218 267 Osculating, hyperquadric, Darboux 217 218 220 266 267 Osculating, plane of Monge equation 281 Osculating, plane of order q of curve 48 Osculating, subspace of 2-submanifold 288—290 Osculating, subspace of curve 53 Osculating, subspace of Grassmannian 50 Osculating, subspace of hypersurface 54 Osculating, subspace of image of Grassmannian 52 Osculating, subspace of order q to submanifold 46 48 151 Osculating, subspace, 1st of submanifold 41 Osculating, subspace, 2nd of 119 120 124 127 128 Osculating, subspace, 2nd of Segre variety 41 Osculating, subspace, 2nd of submanifold 41 42 47 55 56 62 82 85 145 147—150 152 165 167 Osculating, subspace, 2nd of Veronese variety 42 Osculating, subspace, 3rd 45 Osculating, subspace, sequence of 46 154 Ostianu 199 207 316 320 321 p-form, differential 207 p-form, exterior 8 9 p-form, exterior, value of 9 p-web 206 Pair of conjugate directions, double 68 Pair of conjugate directions, simple 68 Pair of conjugate directions, threefold 66 Pair of hypersurfaces 272 Pair of hypersurfaces, moving frame of 272 Pair, autodual 255 Pair, one-sided stratifiable 187 188 206 Parabolic point 54 61 133 Parabolic point of 2-dimensional submanifold 67 Parabolic submanifold 132—135 Parabolic submanifold, 3-dimensional of rank 2 141 Parabolic submanifold, complete 134 141 Parabolic surface 135 Parallel 2-planes 21 Parallel 2-planes, one-parameter family of 2 60 Parallel 2nd fundamental form vi 221 222 Parallel straight lines 21 Parallel subbundle 205 Parallelism of 2nd fundamental form 221 Parameters, homogeneous 29 34 Parameters, nonhomogeneous 34 Parameters, principal 6 263 parameters, secondary 6 147 260 263 283 294 Parametrization 270 Pedoe 311 Pencil of 1st normals 231 250 Pencil of 1st normals, canonical classical 250 267 Pencil of 2nd fundamental forms 57 Pencil of 2nd normals 231 Pencil of conies 57 Pencil of Darboux quadrics, classical 249 Pencil of hyperplanes 282 Pencil of normalizations 231 Pencil of osculating Darboux hyper quadrics 218 223 230 231 235 237 249 267 Pencil of planes 128 Pencil of straight lines 137 275 Pencil of subspaces 28 Peterson submanifold 104 112 Peterson submanifold, generalized 107 109 111 Peterson submanifold, generalized, construction of 108 Peterson submanifold, generalized, existence of 108 Pfaffian, equation 281 Pfaffian, equations, system of, completely integrable 11 Pfaffian, equations, system of, in involution 12 31 87 89 104 105 108 126 139 141 153 191 253 Pfaffian, form 9 279 Pfaffian, form, closed 10 Pfaffian, form, exact 10 Planar element, stationary subgroup of 35 Planar hyperband 259 262 263 266 Planar point 65 Plane(s) 115 118 146 251 Plane(s) at infinity 173 Plane(s), axial 111 Plane(s), characteristic 120 124 128 Plane(s), conisecant 30 31 Plane(s), cubic curve 70 Plane(s), curve 269 Plane(s), curve of 2nd order 64 Plane(s), curve, smooth 270 Plane(s), distribution of 60 143 Plane(s), double tangent 249 Plane(s), family of 147 207 Plane(s), generator of, hypersurface 131 Plane(s), generator of, image of Grassmannian 28 Plane(s), generator of, Segre variety 29 51 Plane(s), generators, family of 117 149 Plane(s), improper 137 Plane(s), invariant 260 264 Plane(s), moving frame of 143 Plane(s), normalizing 260 261 Plane(s), normalizing of hyperband 260 261 Plane(s), normalizing, dual 260 Plane(s), osculating 281 Plane(s), pencil of 128 Plane(s), polar 260 261 Pluecker, coordinates 28 Pluecker, hyperquadric 28 Pohl 296 318 Poincare lemma 10 Point(s) at infinity 137 173 Point(s), 4th harmonic 61 Point(s), axial 55 65 127 Point(s), classification 57 61 63 70 72 Point(s), classification of 64—67 Point(s), classification of 67—70 72 Point(s), double 69 Point(s), elliptic 61 Point(s), elliptic of 2-dimensional submanifold 67 Point(s), focal 117 132 Point(s), frame 129 Point(s), hyperbolic 61 Point(s), hyperbolic of 2-dimensional submanifold 67 Point(s), invariant 219 241 Point(s), nonsingular 33 34 Point(s), nonsingular of general type 54 Point(s), parabolic 54 61 133 Point(s), parabolic of 2-dimensional submanifold 67 Point(s), planar 65 Point(s), regular 7 118 Point(s), simple 69 Point(s), singular 7 36—38 118 132 Point(s), stationary subgroup of 24 43 183 Point(s), stationary subgroup of, invariant form of 24 183 Point(s), submanifold 269 Point(s), unity 18 polar 176 184 193 Polar, hyperplane 260 Polar, plane 260 261 Polar, second 215 Polar-conjugate normals 177 223 230 231 234 Pole, harmonic 205 Polovtseva 67 70 72 321 Polynomial, algebraic homogeneous 195 279 Popov 268 321 322 Positive definite, 2nd fundamental tensor of hypersurface 226 242 Positive definite, hyperquadric 23 Principal form 184 Principal parameters 6 263 Product, direct 30
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