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M.A.Akivis, V.V.Goldberg — Projective Differential Geometry of Submanifolds
M.A.Akivis, V.V.Goldberg — Projective Differential Geometry of Submanifolds



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Название: Projective Differential Geometry of Submanifolds

Авторы: 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.

Graduate students majoring in differential geometry will find this monograph of great interest, as will researchers in differential and algebraic geometry, complex analysis and theory of several complex variables.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 362

Добавлена в каталог: 11.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Submanifold(s), 2-dimensional      59 64—66 158 207 289
Submanifold(s), 2-dimensional of 4th order      30
Submanifold(s), 2-dimensional, 2nd fundamental form of      59—61 64—66 158 289
Submanifold(s), 2-dimensional, asymptotic direction of      60—61 64—66
Submanifold(s), 2-dimensional, conjugate net of      60
Submanifold(s), 2-dimensional, infinitesimal displacement of moving frame of      287
Submanifold(s), 2-dimensional, invariant normalization of      207
Submanifold(s), 2-dimensional, moving frame of      287
Submanifold(s), 2-dimensional, osculating subspace of      289 290
Submanifold(s), 2-dimensional, semi-asymptotic net of      61
Submanifold(s), 2-dimensional, smooth      285
Submanifold(s), 2-dimensional, tangent subspace to      290
Submanifold(s), 2-dimensional, tangentially degenerate      67
Submanifold(s), 2-dimensional, Veronese mapping of      64—66
Submanifold(s), 2-ruled      72
Submanifold(s), 2nd fundamental form of      40 42 47 62 71 165 169 277
Submanifold(s), 2nd fundamental form of, 2-dimensional      59—61 64—66
Submanifold(s), 2nd fundamental form of, 3-dimensional      56—58 67—70
Submanifold(s), 2nd fundamental form of, carrying conjugate net      73
Submanifold(s), 2nd fundamental tensor of      39 73 148 149 152 165 269 277
Submanifold(s), 2nd fundamental tensor of, rank of      56
Submanifold(s), 2nd osculating subspace of      41 62
Submanifold(s), 3-dimensional      56—58 72
Submanifold(s), 3-dimensional with 6-dimensional osculating sub-space      72
Submanifold(s), 3-dimensional with complete system of one-dimensional asymptotic distributions      151
Submanifold(s), 3-dimensional, 2nd fundamental form of      56—61 64—66
Submanifold(s), 3-dimensional, asymptotic direction of      57—58 67—70
Submanifold(s), 3-dimensional, carrying net of asymptotic lines      151
Submanifold(s), 3-dimensional, conjugate directions of      57—58 67—70
Submanifold(s), 3-dimensional, conjugate net of      57
Submanifold(s), 4th order neighborhood of      199
Submanifold(s), 5-dimensional      72
Submanifold(s), bundle of 1st order frames of      93
Submanifold(s), carrying complete system of conjugate directions      165
Submanifold(s), carrying conjugate net      73 85 99 104 111 188 190 200 207
Submanifold(s), carrying conjugate net, 2nd fundamental form of      62 73 165
Submanifold(s), carrying conjugate net, existence of      85
Submanifold(s), carrying conjugate net, invariant normalization of      199—205 207
Submanifold(s), carrying conjugate net, quasi-Laplace transformation of      100 112
Submanifold(s), carrying family of rectilinear generators      171
Submanifold(s), carrying m-conjugate system      101
Submanifold(s), carrying net      207
Submanifold(s), complete parabolic      134 141
Submanifold(s), complete regular      137
Submanifold(s), conic      104 112 168
Submanifold(s), conic, 2-dimensional      109 110
Submanifold(s), conic, equations of      104 110
Submanifold(s), conic, existence of      105
Submanifold(s), conic, generalized      107—110
Submanifold(s), conic, vertex of      168
Submanifold(s), determinant      29 30 67 130 131
Submanifold(s), determinant, algebraic      53
Submanifold(s), doubly foliated      83
Submanifold(s), equipped      174
Submanifold(s), filtration of      62 67
Submanifold(s), generator of      116—118 132 134 148 150
Submanifold(s), infinitesimal displacement of frame of      74
Submanifold(s), integral      144 170
Submanifold(s), moving frame of      34
Submanifold(s), normalized      174 205 206 266
Submanifold(s), normalized of codimension 2      205
Submanifold(s), normalized, element of      174
Submanifold(s), normalized, infinitesimal displacement of frame of      174 282
Submanifold(s), normalized, tangentially degenerate      205
Submanifold(s), parabolic      132—135
Submanifold(s), parabolic, 3-dimensional of rank 2      14
Submanifold(s), parabolic, complete      134 141
Submanifold(s), Peterson      104 112
Submanifold(s), Peterson, generalized      108 109 111
Submanifold(s), point      269
Submanifold(s), quasiasymptotic      71 111
Submanifold(s), regular      134 137
Submanifold(s), regular, complete      137
Submanifold(s), Riemannian      133
Submanifold(s), ruled      94 171
Submanifold(s), shadow      112 170
Submanifold(s), singular      269
Submanifold(s), smooth      285
Submanifold(s), support      255 261
Submanifold(s), support, plane generator of      255 257 260 261
Submanifold(s), system of      277 278
Submanifold(s), system of, algebraic      277 278
Submanifold(s), tangent subspace to      34 36 62 145 174
Submanifold(s), tangentially degenerate      38 96 103 113 116—118 121 122 124 126 129 141 166
Submanifold(s), tangentially degenerate of codimension 3      101
Submanifold(s), tangentially degenerate of rank 2      118
Submanifold(s), tangentially degenerate of rank one      116 118 122 123
Submanifold(s), tangentially degenerate, 2nd osculating subspace of      119 120 124 127 128
Submanifold(s), tangentially degenerate, 3-dimensional      57 58 67—70
Submanifold(s), tangentially degenerate, 3-dimensional of rank 2      137
Submanifold(s), tangentially degenerate, autodual      117 255
Submanifold(s), tangentially degenerate, carrying holonomic focal net      125
Submanifold(s), tangentially degenerate, generator of      116 118 132 134 148 150
Submanifold(s), tangentially degenerate, infinitesimal displacement of frame of      114
Submanifold(s), tangentially degenerate, normalized      205
Submanifold(s), tangentially degenerate, self-dual      117 255
Submanifold(s), tangentially degenerate, tank of      96 166
Submanifold(s), tangentially degenerate, without singularities      118
Submanifold(s), tangentially nondegenerate      38 73 75 103 113 116
Submanifold(s), with special projective structure      vi
Subnormal of 1st normal      174 175 193
Subnormal of 1st normal of hyperquadric      177
Subspace(s)      169
Subspace(s), (r-1)-secant      29 30
Subspace(s), 1st normal of      36 40 53 61 147 167
Subspace(s), asymptotic      145 147
Subspace(s), characteristic      49 52 53 120
Subspace(s), conjugate      215
Subspace(s), conjugate, complementary      44 277
Subspace(s), differential equations of      143
Subspace(s), infinitesimal displacement of frame of      26 143
Subspace(s), invariant      147
Subspace(s), linear      v
Subspace(s), normal      36 42 43 45 61 62 147
Subspace(s), normal, basis of      43
Subspace(s), normal, dimension of      43
Subspace(s), normal, reduced      45 81
Subspace(s), osculating of 2-submanifold      288—290
Subspace(s), osculating of curve      53
Subspace(s), osculating of Grassmannian      50
Subspace(s), osculating of hypersurface      54
Subspace(s), osculating of order k of image of Grassmannian      52
Subspace(s), osculating of order q to submanifold      46 48 151
Subspace(s), osculating, 1st of submanifold      41
Subspace(s), osculating, 2nd of $V^{m}_{r}$      119 120 124 127 128
Subspace(s), osculating, 2nd of Segre variety      41
Subspace(s), osculating, 2nd of submanifold      41 42 47 55 56 62 82 85 119—128 145 147—150 152 165 167
Subspace(s), osculating, 2nd of Veronese variety      42
Subspace(s), osculating, 3rd      45
Subspace(s), pencil of      28
Subspace(s), projective      116 143
Subspace(s), stationary subgroup of      25 26
Subspace(s), stationary subgroup of, invariant form of      26
Subspace(s), tangent      5 34 35 61 145
Subspace(s), tangent, to 2-submanifold      290
Subspace(s), tangent, to curve      53
Subspace(s), tangent, to hypersurface $V^{4}\subset P^{5}$      291
Subspace(s), tangent, to image of Grassmannian      35
Subspace(s), tangent, to Segre variety      37
Subspace(s), tangent, to submanifold      34 35 61 145
Subspace(s), tangent, to tangentially degenerate submanifold      115
Subspace(s), tangent, to Veronese variety      138
Summation      1
Support submanifold      255 261
Support submanifold, plane generator of      255 257 260 261
Surface(s)      90 91 173 209 245
Surface(s), 2nd fundamental form of      91 92
Surface(s), asymptotic line of      251
Surface(s), canonical tangent of      250 251
Surface(s), conic      108
Surface(s), conjugate net on      55 91 92
Surface(s), Darboux line of      251
Surface(s), degeneracy of      251
Surface(s), Fubini linear element of      214 251
Surface(s), Fubini net of      251
Surface(s), invariant of      245
Surface(s), Laplace transform of      90 91 111
Surface(s), Laplace transform of, 2nd fundamental form of      98
Surface(s), moving frame of      90 91 245 252
Surface(s), parabolic      135
Surface(s), projective theory of      267
Surface(s), projectively deformable      267
Surface(s), ruled      29 156 158 235 251
Surface(s), ruled of 2nd order      30
Surface(s), with common Lie quadrics      252 253
Surjective mapping      7
svec      v 266 267 326
Svoboda      171 327
Symmetric double transformation      62
Symmetric embedding      30
Symmetric matrix      113 138
Symmetric tensor      130 177 232 243
Symmetric tensor, nondegenerate      177
Symmetric tensor, product      37
Symmetrization      27
Symmetroid, cubic      31 67—69 286 287
System of 2nd fundamental forms      169
System of 2nd fundamental forms of $V^{2}\subset P^{n}$      64—67
System of 2nd fundamental forms of $V^{3}\subset P^{n}$      67—70
System of hypersurfaces      271
System of Pfaffian equations in involution      12 31 87 89 105 108 126 139 141 153 191 253
System of Pfaffian equations, completely integrable      11
System of points, rank of      81 83 201
System of submanifolds      277 278
System of submanifolds, algebraizable      277 278
System of tensors      240
System, closed      109
System, complete 2-component      148
System, complete 2-component, doubly foliated      149 150
System, complete 2-component, integrable conjugate      169
System, complete multicomponent      150
System, completely integrable      11
System, conic      105 107 205
System, conjugate      78—80
System, conjugate of distributions      165
System, conjugate, 2-component      112 148 165 169—171
System, conjugate, conic      205
System, conjugate, conic 2-component      169
System, conjugate, multicomponent      171
System, m-conjugate      78—80 82 83 98—101 104 106
System, m-conjugate, 2nd Laplace transform of      101
System, m-conjugate, conic      105 108 205
System, m-conjugate, existence of      85
System, m-conjugate, Laplace transform of      100 111
System, m-conjugate, R      112
T-pair of complexes      112
Tangency of 1st order      216
Tangency of 2nd order      211 216
Tangency of 3rd order      217 218 220
Tangency, double      69
Tangent bundle      5 178 183
Tangent bundle, affine connection in      179 183 193
Tangent bundle, element of      5
Tangent canonical      250 251
Tangent hyperplane      49 136 284
Tangent hyperplane, to hypercone      284
Tangent hyperplane, to hyperquadric      219
Tangent hyperplane, to hypersurface      211 216 272
Tangent plane, double      249
Tangent space      5 19
Tangent space, basis of      19
Tangent subspace to 2-dimensional submanifold      290
Tangent subspace to curve      53
Tangent subspace to hypersurface $V^{4}\subset P^{5}$      291
Tangent subspace to image of Grassmannian      35
Tangent subspace to Segre variety      37
Tangent subspace to submanifold      34 35 61 145
Tangent subspace to tangentially degenerate submanifold      115
Tangent subspace to Veronese variety      138
Tangent vector      5
Tangent vector, coordinates of      14 48
Tangential coordinates      20 49 261
Tangential frame      20
Tangential frame of hyperband      257
Tangential frame, infinitesimal displacement of      20
Tangentially degenerate hypersurface      256 260
Tangentially degenerate hypersurface of rank one      116
Tangentially degenerate hypersurface of rank two      291
Tangentially degenerate submanifold      38 96 113 116—118 121 122 124 126 129 141 166
Tangentially degenerate submanifold of codimension 3      101
Tangentially degenerate submanifold of rank 2      118
Tangentially degenerate submanifold of rank one      116 118 122 123
Tangentially degenerate submanifold, 2-dimensional      67
Tangentially degenerate submanifold, 2nd osculating subspace of      119 120 124 127 128
Tangentially degenerate submanifold, 3-dimensional      57 58 67—70
Tangentially degenerate submanifold, 3-dimensional of rank 2      137
Tangentially degenerate submanifold, autodual      117 255
Tangentially degenerate submanifold, carrying holonomic focal net      125
Tangentially degenerate submanifold, generator of      116—118 132 134 148 150
Tangentially degenerate submanifold, infinitesimal displacement of frame of      114
Tangentially degenerate submanifold, normalised      205
Tangentially degenerate submanifold, rank of      96 166
Tangentially degenerate submanifold, self-dual      117 255
Tangentially degenerate submanifold, tangent subspace to      115
Tangentially degenerate submanifold, without singularities      118
Tangentially degenerate surface      67
Tangentially nondegenerate hypersurface      222
Tangentially nondegenerate submanifold      38 73 75 103 113 116
Teixidor      271 327
Tensor(s)      176 197
Tensor(s) of projective curvature      180 181
Tensor(s), 1st canonical      230
Tensor(s), 2nd fundamental of, hypersurface      54 218 221
Tensor(s), 2nd fundamental of, submanifold      39 73 148 149 152 165 269 277
Tensor(s), 3rd fundamental      44 152
Tensor(s), apolar      233
Tensor(s), curvature      17 179
Tensor(s), Darboux      112 171 213 218 218—220 222 225—228 244 266 269
Tensor(s), Darboux of hyperband      265
Tensor(s), Darboux, degenerate      227
Tensor(s), Darboux, nondegenerate      236
Tensor(s), determinant of      236
Tensor(s), differential equations of      3
Tensor(s), field      vii 6
Tensor(s), fundamental of order q      48
Tensor(s), inverse      178 199 203 211 214 227 258
Tensor(s), law of transformation of      4 115
Tensor(s), mixed      39
Tensor(s), mutually inverse      203 214 258
Tensor(s), non singular      197
Tensor(s), nondegenerate      257 258
Tensor(s), nondegenerate, Darboux      226
Tensor(s), nondegenerate, symmetric      177
Tensor(s), rank of      295
Tensor(s), relative      4 212 213 242 246 259
Tensor(s), Ricci      181 185 186
Tensor(s), Ricci, alternated      185
Tensor(s), symmetric      130 232 233 243
Tensor(s), symmetric, nondegenerate      177
Tensor(s), symmetric, product      37
Tensor(s), system of      240
Tensor(s), torsion      17 179
Tensor(s), trace of      225
Tensorial method      vii
Tensorial square      30
Terentjeva      268 327
Terrachini      71
Theorem, Abel      270 278 295
Theorem, Abel, generalization of      295
Theorem, Euler      195
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