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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
$\mu$-ruled submanifold      72
(0, 2)-tensor      214 215 232
(0, 3)-tensor      215
(r-1)-secant subspace      30
1-form, differential      8
1st canonical tensor      230
1st generalized Reiss theorem      272
1st integral      15 26 116
1st neighborhood of hyperband      257
1st normal(s)      174 250
1st normal(s) of hyperband      262
1st normal(s) of hypersurface      212 222 241 242 248
1st normal(s), invariant      230
1st normal(s), pencil of      231 250
1st normal(s), space of hypersurface      53
1st normal(s), subnormal of      174 175 193
1st normal(s), subspace      36 40 53—55 61 84 147 167
1st normal(s), subspace reduced      42 54 55 81 167 206
1st order frames      34
1st order frames, bundle of      93
1st order tangency      216
1st osculating subspace of submanifold      41
2-component conjugate system      165 169—171
2-component conjugate system, conic      169
2-component system of asymptotic distributions      148
2-component system of asymptotic distributions, complete      148
2-component system of asymptotic distributions, complete, doubly foliated      149 150
2-component system of asymptotic distributions, complete, integrable conjugate      169
2-dimensional conic submanifold      109 110
2-dimensional distribution      77
2-dimensional Grassmann 3-web      276
2-dimensional submanifold      59 64—66 158 207 289
2-dimensional submanifold of 4th order      30
2-dimensional submanifold, 2nd fundamental form of      59—61 64—66 158 289
2-dimensional submanifold, asymptotic direction of      60—61 64—66
2-dimensional submanifold, conjugate net of      60
2-dimensional submanifold, infinitesimal displacement of moving frame of      287
2-dimensional submanifold, invariant normalization of      207
2-dimensional submanifold, moving frame of      287
2-dimensional submanifold, osculating subspace of      289 290
2-dimensional submanifold, semi-asymptotic net of      61
2-dimensional submanifold, smooth      285
2-dimensional submanifold, tangent subspace to      290
2-dimensional submanifold, tangentially degenerate      67
2-dimensional submanifold, tangentially degenerate normalized      205
2-dimensional submanifold, Veronese mapping of      64—66
2-dimensional Veronese variety      67 285 290 295
2-plane(s)      21 59 70 137 272 291
2-plane(s), one-parameter family of      60
2-ruled submanifold      72
2-web      77
2-web, curvilinear      77
2-web, quadrilateral      77
2nd fundamental form(s) of conjugate net      74
2nd fundamental form(s) of hyperband      258
2nd fundamental form(s) of hypersurface      53 210 212 244 272—275 277 278 291
2nd fundamental form(s) of image of the Grassmannian      51
2nd fundamental form(s) of Laplace transform      98
2nd fundamental form(s) of Segre variety      41 158
2nd fundamental form(s) of submanifold      40 42 47 62 71 165 169 277
2nd fundamental form(s) of submanifold, 2-dimensional      59—61 64—66 289
2nd fundamental form(s) of submanifold, 3-dimensional      56—58 67—70
2nd fundamental form(s) of submanifold, carrying conjugate net      73
2nd fundamental form(s) of surface      91 92
2nd fundamental form(s) of Veronese variety      41 287
2nd fundamental form(s), parallel      vi 221 222
2nd fundamental form(s), parallelism of      221
2nd fundamental form(s), pencil of      57
2nd fundamental form(s), system of      64—70 169
2nd fundamental tensor of hypersurface      54 218 221
2nd fundamental tensor of hypersurface, covariant differential of      221
2nd fundamental tensor of hypersurface, indefinite      226
2nd fundamental tensor of hypersurface, parallelism of      221
2nd fundamental tensor of hypersurface, positive definite      226 242
2nd fundamental tensor of hypersurface, rank of      54
2nd fundamental tensor of submanifold      39 73 148 149 152 165 269 277
2nd fundamental tensor of submanifold, rank of      56
2nd generalized Reiss theorem      277
2nd Laplace transform      101
2nd neighborhood of hypersurface      233
2nd normal(s)      174
2nd normal(s) of hyperband      262
2nd normal(s) of hyperguadric      177
2nd normal(s) of hypersurface      212 222 224 241 242 248
2nd normal(s), invariant      230
2nd normal(s), pencil of      231
2nd normal(s), subspace      42 45
2nd normal(s), subspace, basis of      43
2nd normal(s), subspace, dimension of      43
2nd normal(s), subspace, reduced      45 81
2nd order, asymptotic direction      48 57 63 71 92 94 145
2nd order, asymptotic direction of 2-submanifold      60—61 64—66
2nd order, asymptotic direction of 3-submanifold      57—58 67—70
2nd order, asymptotic direction on hypersurface      230
2nd order, cone      215
2nd order, conjugate directions      49 57 63
2nd order, curve      31 269
2nd order, frame      42
2nd order, neighborhood of hyperband      257 259 260
2nd order, tangency      211 216
2nd osculating subspace      41 42 47 55 56 61 62 82 85 145 147—150 152 165 167
2nd osculating subspace of $V^{m}_{r}$      119 120 124 127 128
2nd osculating subspace of conjugate net      75
2nd osculating subspace of image of Grassmannian      51
2nd osculating subspace of Segre variety      41
2nd osculating subspace of Veronese variety      42
2nd polar      215
3-conjugate system R      111
3-dimensional projective space      209
3-dimensional Riemannian space of constant curvature      135
3-dimensional submanifold      56—58 72
3-dimensional submanifold, 2nd fundamental form of      56—61
3-dimensional submanifold, asymptotic direction of      57—58 67—70
3-dimensional submanifold, carrying net of asymptotic lines      151
3-dimensional submanifold, conjugate directions of      57—58 67—70
3-dimensional submanifold, conjugate net of      57
3-dimensional submanifold, with 6-dimensional osculating subspace      72
3-dimensional submanifold, with complete system of one-dimensional asymptotic distributions      151
3-plane      56 156
3-web, Grassmann      275
3-web, Grassmann, 2-dimensional      276
3-web, Grassmann, curvature tensor of      275
3-web, Grassmann, GW(3,2,1)      275 276
3-web, Grassmann, GW(3,2,r)      275 276
3-web, Grassmann, hexagonal      275 276
3-web, rectilinear      275
3-web, rectilinear, hexagonal      276
3rd class curve      276
3rd fundamental form      44 49 153
3rd fundamental form of submanifold with conjugate net      75
3rd fundamental tensor      44 152
3rd order conjugate directions      50 71 87 111
3rd order curve      276
3rd order frame      238 241
3rd order frame subbundle      235 237
3rd order neighborhood of hyperband      260 264
3rd order neighborhood of hypersurface      211 225 227 233
3rd order tangency      217 218 220
3rd osculating subspace      45 154
4th harmonic point      61
4th neighborhood of hyperband      262 265
4th neighborhood of hypersurface      212 213 224 233 242
4th neighborhood of submanifold      199
4th order curve      30
4th order curve, 2-submanifold      30
4th order curve, frame      240—242
4th order curve, fundamental object      242
4th order curve, subbundle      237
5-dimensional submanifold      72
5th neighborhood of hypersurface      232 233 241 242
5th order fundamental object      241 244
Abel      270 295 297
Abel, condition      279
Abel, theorem      270 278 295
Abel, theorem, generalization of      295
Absolute of $S^{n}$      134
Absolute, differential invariant      214 233
Absolute, invariant      4 227 231 250
Absolutely invariant form      233 250
Admissible transformation      61 35 39 74 174 179 183 185 200 212
Affine, connection      16 17 184
Affine, connection in normal bundle      183
Affine, connection in tangent bundle      179 183 193
Affine, connection on hypersurface      221
Affine, connection, curvature tensor of      17 179
Affine, connection, torsion tensor of      17 179
Affine, connection, torsion-free      17
Affine, coordinate system      276
Affine, frame      21 22
Affine, geometry      vi 221
Affine, normal      vi 224 268
Affine, normalization      177 180 185
Affine, space      21 173 206 224 266
Affine, space, infinitesimal displacement of frame of      21
Affine, space, structure equations of      22
Affine, space, transformation of      21
Affinor      242
Akivis      61 70 72 82 101 111 112 141 169 171 172 205 207 212 277 282 296—300
Algebraic, cone      130
Algebraic, curve      270
Algebraic, equation      vi 94 95 99 204
Algebraic, family of hyperplanes      119
Algebraic, geometry      v
Algebraic, homogeneous polynomial      195
Algebraic, hypersurface      122 272 278
Algebraic, manifold      269
Algebraic, variety      v 26 117 129 261
Algebraic, variety, projective      278
Algebraizability condition      269
Algebraizability condition for hyperdistribution      282
Algebraizable Grassmann d-web      276
Algebraization problem      269 278 279
Algebraization problem for multidimensional webs      296
Algebraization problem for submanifolds      296
Alternated Ricci tensor      185
Alternation      27
Analytic function      7
Analytic manifold      5
Anti-commutative      8
Apolar tensors      214 233
Apolarity      213—216 228 233 259 265
Arbitrariness of general integral element      13
Asymptotic, cone(s)      47
Asymptotic, cone(s) of 2nd order of hypersurface      210
Asymptotic, cone(s) of image of Grassmannian      51 53
Asymptotic, cone(s) of order q      48
Asymptotic, cone(s), filtration of      49 53
Asymptotic, curve      47 54 145 171
Asymptotic, curve of 2nd order      48
Asymptotic, curve of hyperdistribution      281 282
Asymptotic, curve of hypersurface      211
Asymptotic, curve of manifold of hypercones      284
Asymptotic, curve of Monge equation      281
Asymptotic, curve of order q      48 71
Asymptotic, curve of surface      251
Asymptotic, curve, rectilinear      155 158 159
Asymptotic, direction of 2-submanifold      60—61 64—66
Asymptotic, direction of 2nd order      48 57 63 71 92 94 145
Asymptotic, direction of 3-submanifold      57—58 67—70
Asymptotic, direction of image of Grassmannian      53
Asymptotic, direction of order q      48 71
Asymptotic, direction on hypersurface      230
Asymptotic, direction, double      66 70
Asymptotic, direction, real      67
Asymptotic, direction, threefold      69
Asymptotic, distribution      145 171
Asymptotic, distribution, complete 2-component      148
Asymptotic, distribution, complete multicomponent      150 171
Asymptotic, distribution, integrable      145
Asymptotic, distribution, one-dimensional      161
Asymptotic, holonomic net      155
Asymptotic, hypercone      279—284
Asymptotic, subspace      145 147
Atanasyan      205—207 300
AutoDual      117 255
Autodual, hyperband      255 256
Autodual, pair      255
Autodual, tangentially degenerate submanifold      117 255
Autopolar simplex      23
Axial, normalisation      205
Axial, plane      111
Axial, point      55 65 127
Axis of Koenigs net      170
Baecklund      295 301
Baimuratov      57 72 171 296 300 301
Balazyuk      321
Baldasari      72 301
Band      268
Barner      vii 171 301
Base of bundle of 1st order frames      35
Base of connection      178
Base of normal bundle      182
Base of subbundle      24
Base, form      14 35
Base, natural      14
Basis of 2nd normal subspace      43
Basis of cotangent space      6 19
Basis of hyperplane      131
Basis of tangent space      19
Basis of vector space      1 35
Basis, dual      5
Basis, form      35
Basis, form of Grassmannian      38 128
Basis, form of hypersurface      210 244
Bazylev      82 111 112 206 301 302
Bell      111 302
Berezina      207 302
Bertini      302
Bijective mapping      7
Bilinear equation      223
Bilinear form      49 132
Bisecant      29
Bisecant, variety      29 31 285 286
Bisecant, variety, degenerate      285 286
BLANK      112 302
Blaschke      72 259 268 302 303
Blaschke affine normal      vi 224 268
Bol      v 72 209 303
Bompiani      71 72 296 303
Borisenko      134 141 303
Bortolotti      v 266 303
Botsu      275 295 303
Brauner      v 141 304
Bryant      304
Bubyakin      70 304
Bundle of 1st order frames      35 93
Bundle of 1st order frames, base of      35
Bundle of 1st order frames, typical fiber of      35
Bundle of 2nd fundamental forms      40
Bundle of 2nd fundamental forms, dimension of      40
Bundle of 3rd fundamental forms      44
Bundle of 3rd fundamental forms, dimension of      45
Bundle of hyperplanes      121 122 127 189
Bundle of hyperplanes, center of      121 122 189
Bundle of planes      176
Bundle of straight lines      275
Bundle of subspaces      28
Bundle, cotangent      19
Bundle, frame      5 16
Bundle, frame, dimension of      5
1 2 3 4 5 6 7 8 9
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