The geometry of curvature homogeneous pseudo-Riemannian manifolds:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
London
Imperial College Press
2007
|
| Schriftenreihe: | ICP advanced texts in mathematics
2 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016222466&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XII, 376 S. |
| ISBN: | 9781860947858 1860947859 |
Internformat
MARC
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| 020 | |a 9781860947858 |9 978-1-86094-785-8 | ||
| 020 | |a 1860947859 |9 1-86094-785-9 | ||
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| 100 | 1 | |a Gilkey, Peter B. |d 1946- |e Verfasser |0 (DE-588)1024266850 |4 aut | |
| 245 | 1 | 0 | |a The geometry of curvature homogeneous pseudo-Riemannian manifolds |c Peter B. Gilkey |
| 264 | 1 | |a London |b Imperial College Press |c 2007 | |
| 300 | |a XII, 376 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a ICP advanced texts in mathematics |v 2 | |
| 650 | 4 | |a Curvature | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Riemannian manifolds | |
| 650 | 0 | 7 | |a Pseudo-Riemannscher Raum |0 (DE-588)4176163-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Pseudo-Riemannscher Raum |0 (DE-588)4176163-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 830 | 0 | |a ICP advanced texts in mathematics |v 2 |w (DE-604)BV023102173 |9 2 | |
| 856 | 4 | 2 | |m Digitalisierung UB Augsburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016222466&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016222466 | |
Datensatz im Suchindex
| _version_ | 1819292518644187136 |
|---|---|
| adam_text | Contents
Preface
The Geometry of the Riemann Curvature Tensor
1
1.1
Introduction
........................... 1
1.2
Basic Geometrical Notions
................... 4
1.2.1
Vector spaces with symmetric inner products
.... 4
1.2.2
Vector bundles, connections, and curvature
..... 6
1.2.3
Holonomy and parallel translation
.......... 10
1.2.4
Affine
manifolds, geodesies, and completeness
.... 11
1.2.5
Pseudo-
Riemannian manifolds
............. 12
1.2.6
Scalar Weyl invariants
................. 15
1.3
Algebraic Curvature Tensors and Homogeneity
....... 16
1.3.1
Algebraic curvature tensors
.............. 17
1.3.2
Canonical curvature tensors
.............. 21
1.3.3
The Weyl
conformai
curvature tensor
......... 23
1.3.4
Models
.......................... 24
1.3.5
Various notions of homogeneity
............ 26
1.3.6
Killing vector fields
................... 27
1.3.7 Nilpotent
curvature
................... 28
1.4
Curvature Homogeneity
-
a Brief Literature Survey
..... 28
1.4.1
Scalar Weyl invariants in the Riemannian setting
. . 28
1.4.2
Relating curvature homogeneity and homogeneity
. . 29
1.4.3
Manifolds modeled on symmetric spaces
....... 30
1.4.4
Historical survey
.................... 31
1.5
Results from Linear Algebra
.................. 32
1.5.1
Symmetric and anti-symmetric operators
....... 32
The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds
2.
1.5.2
The spectrum of an operator
............. 32
1.5.3
Jordan normal form
.................. 33
1.5.4
Self-adjoint maps in the higher signature setting
... 34
1.5.5
Technical results concerning differential equations
. . 35
1.6
Results from Differential Geometry
.............. 38
1.6.1
Principle bundles
.................... 39
1.6.2
Geometric readability
................. 39
1.6.3
The canonical algebraic curvature tensors
...... 41
1.6.4
Complex geometry
................... 47
1.6.5
Rank l-symmetric spaces
............... 51
1.6.6
Conformai
complex space forms
............ 53
1.6.7 Kahler
geometry
.................... 54
1.7
The Geometry of the Jacobi Operator
............ 54
1.7.1
The Jacobi operator
.................. 55
1.7.2
The higher order Jacobi operator
........... 57
1.7.3
The
conformai
Jacobi operator
............ 59
1.7.4
The complex Jacobi operator
............. 60
1.8
The Geometry of the Curvature Operator
.......... 62
1.8.1
The skew-symmetric curvature operator
....... 62
1.8.2
The
conformai
skew-symmetric curvature operator
. 65
1.8.3
The Stanilov operator
................. 66
1.8.4
The complex skew-symmetric curvature operator
. . 66
1.8.5
The
Szabó
operator
................... 68
1.9
Spectral Geometry of the Curvature Tensor
......... 69
1.9.1
Analytic continuation
.................. 70
1.9.2
Duality
.......................... 72
1.9.3
Bounded spectrum
................... 75
1.9.4
The Jacobi operator
.................. 78
1.9.5
The higher order Jacobi operator
........... 81
1.9.6
The
conformai
and complex Jacobi operators
.... 82
1.9.7
The Stanilov and the
Szabó
operators
........ 83
1.9.8
The skew-symmetric curvature operator
....... 84
1.9.9
The
conformai
skew-symmetric curvature operator
. 86
Curvature Homogeneous Generalized Plane Wave Manifolds
87
2.1
Introduction
........................... 87
2.2
Generalized Plane Wave Manifolds
.............. 90
2.2.1
The geodesic structure
................. 92
2.2.2
The curvature tensor
.................. 93
Contents
3.
2.2.3
The geometry of the curvature tensor
......... 94
2.2.4
Local scalar invariants
................. 94
2.2.5
Parallel vector fields and holonomy
.......... 96
2.2.6
Jacobi vector fields
................... 96
2.2.7
Isometries
........................ 97
2.2.8
Symmetric spaces
.................... 99
2.3
Manifolds of Signature
(2,2).................. 101
2.3.1
Immersions as hypersurfaces in flat space
....... 103
2.3.2
Spectral properties of the curvature tensor
...... 105
2.3.3
A complete system of invariants
............ 107
2.3.4
Isometries
........................ 109
2.3.5
Estimating
кРіЯ
if min(p, q)
— 2............ 114
2.4
Manifolds of Signature
(2,4).................. 115
2.5
Plane Wave Hypersurfaces of Neutral Signature (p,p)
. . . 119
2.5.1
Spectral properties of the curvature tensor
...... 123
2.5.2
Curvature homogeneity
................. 128
2.6
Plane Wave Manifolds with Flat Factors
........... 130
2.7
Nikčević
Manifolds
....................... 135
2.7.1
The curvature tensor
.................. 137
2.7.2
Curvature homogeneity
................. 139
2.7.3
Local isometry invariants
................ 141
2.7.4
The spectral geometry of the curvature tensor
.... 145
2.8
Dunn Manifolds
......................... 149
2.8.1
Models and the structure groups
........... 151
2.8.2
Invariants which are not of Weyl type
........ 155
2.9
fc-Curvature Homogeneous Manifolds I
............ 156
2.9.1
Models
.......................... 159
2.9.2
Affine
invariants
..................... 162
2.9.3
Changing the signature
................. 164
2.9.4
Indecomposability
.................... 165
2.10
fc-Curvature Homogeneous Manifolds II
........... 166
2.10.1
Models
.......................... 168
2.10.2
Isometry groups
..................... 171
Other Pseudo-Riemannian Manifolds
181
3.1
Introduction
........................... 181
3.2
Lorentz
Manifolds
....................... 182
3.2.1
Geodesies and curvature
................ 185
3.2.2
Ricci
blowup
...................... 187
The Geometry of Curvature Homogeneous Pseudo-Riemannian Manifolds
3.2.3
Curvature homogeneity
................. 188
3.3
Signature
(2,2)
Walker Manifolds
............... 193
3.3.1
Osserman curvature tensors of signature
(2,2) .... 194
3.3.2
Indefinite
Kahler
Osserman manifolds
........ 196
3.3.3
Jordan Osserman manifolds which are not
nilpotent 197
3.3.4
Conformally Osserman manifolds
........... 198
3.4
Geodesic Completeness and
Ricci
Blowup
.......... 201
3.4.1
The geodesic equation
................. 201
3.4.2
Conformally Osserman manifolds
........... 202
3.4.3
Jordan Osserman Walker manifolds
.......... 206
3.5
Fiedler Manifolds
........................ 206
3.5.1
Geometric properties of Fiedler manifolds
...... 207
3.5.2
Fiedler manifolds of signature
(2,2).......... 209
3.5.3 Nilpotent
Jacobi manifolds of order 2r
........ 209
3.5.4 Nilpotent
Jacobi manifolds of order
2т
+ 1...... 213
3.5.5
Szabó
nilpotent
manifolds of arbitrarily high order
. 216
4.
The Curvature Tensor
219
4.1
Introduction
........................... 219
4.2
Topological Results
....................... 221
4.2.1
Real vector bundles
................... 221
4.2.2
Bundles over
projective
spaces
............. 222
4.2.3
Clifford algebras in arbitrary signatures
....... 223
4.2.4
Riemannian Clifford algebras
............. 224
4.2.5
Vector fields on spheres
................. 226
4.2.6
Metrics of higher signatures on spheres
........ 226
4.2.7
Equivariant vector fields on spheres
.......... 227
4.2.8
Geometrically symmetric vector bundles
....... 228
4.3
Generators for the Spaces Alg0 and
,41g!........... 229
4.3.1
A lower bound for v{m) and for v (m)
........ 231
4.3.2
Geometric realizability
. . . .
:
............. 233
4.4
Jordan Osserman Algebraic Curvature Tensors
....... 234
4.4.1
Neutral signature Jordan Osserman tensors
..... 235
4.4.2
Rigidity results for Jordan Osserman tensors
..... 238
4.5
The
Szabó
Operator
...................... 241
4.5.1
Szabó 1-models
..................... 242
4.5.2
Balanced
Szabó
pseudo-Riemannian manifolds
.... 243
4.6
Conformai
Geometry
...................... 245
4.6.1
The Weyl model
.................... 245
Contents
4.6.2
Conformally Jordan Osserman 0-models
....... 246
4.6.3
Conformally Osserman 4-dimensional manifolds
. . . 247
4.6.4
Conformally Jordan Ivanov-Petrova 0-models
.... 249
4.7
Stanilov Models
......................... 251
4.8
Complex Geometry
....................... 253
5.
Complex Osserman Algebraic Curvature Tensors
257
5.1
Introduction
........................... 257
5.1.1
Clifford families
..................... 257
5.1.2
Complex Osserman tensors
.............. 258
5.1.3
Classification results in the algebraic setting
..... 259
5.1.4
Geometric examples
.................. 260
5.1.5
Chapter outline
..................... 261
5.2
Technical Preliminaries
..................... 261
5.2.1
Criteria for complex Osserman models
........ 262
5.2.2
Controlling the eigenvalue structure
.......... 263
5.2.3
Examples of complex Osserman 0-models
...... 264
5.2.4
Reparametrization of a Clifford family
........ 265
5.2.5
The dual Clifford family
................ 265
5.2.6
Compatible complex models given by Clifford families
266
5.2.7
Linearly independent endomorphisms
......... 269
5.2.8
Technical results concerning Clifford algebras
.... 272
5.3
Clifford Families of Rank
1 .................. 276
5.4
Clifford Families of Rank
2 .................. 278
5.4.1
The tensor CxAj,
+
c2AJs
............... 279
5.4.2
The tensor c0A{.t.) +cxAJl
+
c2AJs
.......... 286
5.5
Clifford Families of Rank
3 .................. 288
5.5.1
Technical results
.................... 288
5.5.2
The tensor A
=
c^Aj,
+
с2Ајѕ + сгАЈз........
291
5.5.3
The tensor A
=
с0А(.г)
+
ciAj,
+
c2AJs
+
сзАј,
.. 292
5.6
Tensors A
=
ciAj,
+ ... +
clAJi for
í
> 4 .......... 295
5.7
Tensors A = c0A{.t.) -f-ciAj, +... + ciAjt for l>
4..... 301
6.
Stanilov-Tsankov Theory
309
6.1
Introduction
........................... 309
6.1.1
Jacobi Tsankov manifolds
............... 310
6.1.2
Skew Tsankov manifolds
................. 311
6.1.3
Stanilov-Videv manifolds
............... 312
xii
The Geometry of Curvature Homogeneous
Pseudo-
Riemannian Manifolds
6.1.4
Jacobi Videv manifolds and
0-
models
......... 313
6.2
Riemannian Jacobi Tsankov Manifolds and 0-Models
.... 313
6.2.1
Riemannian Jacobi Tsankov 0-models
........ 314
6.2.2
Riemannian orthogonally Jacobi Tsankov 0-models
. 315
6.2.3
Riemannian Jacobi Tsankov manifolds
........ 322
6.3
Pseudo-Riemannian Jacobi Tsankov 0-Models
........ 323
6.3.1
Jacobi Tsankov 0-models
................ 324
6.3.2
Non
Jacobi Tsankov 0-models with Jl
= 0
V
χ
... 325
6.3.3
0-models with JxJy
=0
Va;, y
Є
V
.......... 326
6.3.4
0-models with AxyAzw
= 0
V x,y,z,w
Є
V
...... 328
6.4
A Jacobi Tsankov
О
-Model
with JxJy
φ
0
for some x,y
. . 331
6.4.1
The model
9Я14
..................... 333
6.4.2
A geometric realization of
9Лн
............ 338
6.4.3
Isometry invariants
................... 340
6.4.4
A symmetric space with model
ЯЯн
......... 343
6.5
Riemannian Skew Tsankov Models and Manifolds
...... 345
6.5.1
Riemannian skew Tsankov models
.......... 347
6.5.2
3-dimensional skew Tsankov manifolds
........ 349
6.5.3
Irreducible 4-dimensional skew Tsankov manifolds
. . 351
6.5.4
Flats in a Riemannian skew Tsankov manifold
.... 353
6.6
Jacobi Videv Models and Manifolds
............. 356
6.6.1
Equivalent properties characterizing Jacobi Videv
models
.......................... 357
6.6.2
Decomposing Jacobi Videv models
.......... 359
Bibliography
361
Index
373
|
| any_adam_object | 1 |
| author | Gilkey, Peter B. 1946- |
| author_GND | (DE-588)1024266850 |
| author_facet | Gilkey, Peter B. 1946- |
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| ctrlnum | (OCoLC)141384863 (DE-599)BVBBV023018342 |
| dewey-full | 516.362 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 |
| dewey-search | 516.362 |
| dewey-sort | 3516.362 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV023018342 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T13:06:43Z |
| institution | BVB |
| isbn | 9781860947858 1860947859 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016222466 |
| oclc_num | 141384863 |
| open_access_boolean | |
| owner | DE-384 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 DE-83 |
| owner_facet | DE-384 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 DE-83 |
| physical | XII, 376 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Imperial College Press |
| record_format | marc |
| series | ICP advanced texts in mathematics |
| series2 | ICP advanced texts in mathematics |
| spellingShingle | Gilkey, Peter B. 1946- The geometry of curvature homogeneous pseudo-Riemannian manifolds ICP advanced texts in mathematics Curvature Geometry, Differential Riemannian manifolds Pseudo-Riemannscher Raum (DE-588)4176163-7 gnd |
| subject_GND | (DE-588)4176163-7 |
| title | The geometry of curvature homogeneous pseudo-Riemannian manifolds |
| title_auth | The geometry of curvature homogeneous pseudo-Riemannian manifolds |
| title_exact_search | The geometry of curvature homogeneous pseudo-Riemannian manifolds |
| title_full | The geometry of curvature homogeneous pseudo-Riemannian manifolds Peter B. Gilkey |
| title_fullStr | The geometry of curvature homogeneous pseudo-Riemannian manifolds Peter B. Gilkey |
| title_full_unstemmed | The geometry of curvature homogeneous pseudo-Riemannian manifolds Peter B. Gilkey |
| title_short | The geometry of curvature homogeneous pseudo-Riemannian manifolds |
| title_sort | the geometry of curvature homogeneous pseudo riemannian manifolds |
| topic | Curvature Geometry, Differential Riemannian manifolds Pseudo-Riemannscher Raum (DE-588)4176163-7 gnd |
| topic_facet | Curvature Geometry, Differential Riemannian manifolds Pseudo-Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016222466&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023102173 |
| work_keys_str_mv | AT gilkeypeterb thegeometryofcurvaturehomogeneouspseudoriemannianmanifolds |