Semiparallel submanifolds in space forms:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
2009
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | VII, 306 S. |
| ISBN: | 9780387499116 9780387499130 |
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| 245 | 1 | 0 | |a Semiparallel submanifolds in space forms |c Ülo Lumiste |
| 264 | 1 | |a New York |b Springer |c 2009 | |
| 300 | |a VII, 306 S. | ||
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Record in the Search Index
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| adam_text |
Contents
0
Introduction
. 1
1
Preliminaries
. 7
1.1
Real Spaces with Bilinear Metric
. 7
1.2
Moving Frames
. 8
1.3
(Pseudo-)Riemannian Manifolds
. 10
1.4
Standard Models of Space and Spacetime Forms
. 11
1.5
Symmetric (Pseudo-)Riemannian Manifolds
. 13
1.6
Semisymmetric (Pseudo-)Riemannian Manifolds
. 16
2
Submanifolds in Space Forms
. 23
2.1
A Submanifold and Its Adapted Frame Bundle
. 23
2.2
Higher-Order Fundamental Forms
. 26
2.3
Fundamental Identities
. 29
2.4
Osculating and Normal Subspaces of Higher Order
. 29
3
Parallel Submanifolds
. 33
3.1
Parallel and
ír-Parallel
Submanifolds
. 33
3.2
Examples:
Segre
and Pliicker Submanifolds
. 36
3.3
Example: Veronese Submanifold
. 40
3.4
Parallel Submanifolds and the Gauss Map
. 43
3.5
Parallel Submanifolds and Local Extrinsic Symmetry
. 44
3.6
Complete Parallel Irreducible Submanifolds as Standard Imbedded
Symmetric
Ќ-Ѕрасеѕ
. 46
4 Semiparallel
Submanifolds
. 51
4.1
The
Semiparallel
Condition and Its Special Cases
. 51
4.2
The
Semiparallel
Condition from the Algebraic Viewpoint
. 54
4.3
Decomposition of
Semiparallel
Fundamental Triplets
. 57
4.4
Triplets of Large Principal Codimension
. 59
4.5 Semiparallel
Submanifolds as Second-Order Envelopes of Parallel
Submanifolds
. 63
Contents
4.6
Second-Order Envelope of
Segre Submanifolds
. 66
4.7
A New Approach to Veronese Submanifolds
. 70
Normally Flat
Semiparallel
Submanifolds
. 73
5.1
Principal Curvature Vectors and the
Semiparallel
Condition
. 73
5.2
Normally Flat Parallel Submanifolds
. 75
5.3
Adapted Frame Bundle for a Second-Order Envelope
. 78
5.4
Second-Order Envelope as Warped Product
. 80
5.5 Semiparallel
Submanifolds of Principal Codimension
1. 84
5.6
Semiparallel Submanifolds of Principal Codimension
2
in
Euclidean Space
. 89
5.7
Normally Flat Semiparallel Submanifolds of Principal
Codimension
2
in Non-Euclidean Space Forms
. 93
Semiparallel Surfaces
. 97
6.1
Semiparallel Spacelike Surfaces
. 97
6.2
The Case of Regular Metrics
. 99
6.3
Veronese Surfaces
. 101
6.4
Second-Order Envelopes of Veronese Surfaces
. 106
6.5
The Case of a Singular Metric
. 108
6.5.1
The subcases where
spanţA,
В}
has singular metric
. 109
6.5.2
The subcases where span{A, B} has regular metric
. 112
6.6
Semiparallel Timelike Surfaces in
Lorentz
Spacetime
Forms
. 114
6.6.1
The principal case
. 115
6.6.2
The exceptional case
. 119
6.7
Spacelike 2-Parallel Surfaces
. 123
6.8
^-Parallel Surfaces as Semiparallel Surfaces
. 130
Semiparallel Three-Dimensional Submanifolds
. 135
7.1
Semiparallel Submanifolds M3 of Principal Codimension m\
< 2. 135
7.2 Nonminimal
Semiparallel M3 of Principal Codimension
т\=Ъ
. 138
7.3
Semiparallel M3 of Principal Codimension m\—A
. 147
7.4
Higher Principal Codimensions: Conclusions
. 154
Decomposition Theorems
. 157
8.
1 Decomposition of Semiparallel Submanifolds
. 157
8.2
Decomposition of Parallel Submanifolds
. 162
8.3
Decomposition of Normally Flat 2-Parallel Submanifolds
. 164
8.4
Structure of Submanifolds with Flat van
der Waerden-Bortolotti
Connection
. 168
Umbilic-Likeness of Main Symmetric Orbits
. 175
9.1
Two Kinds of Symmetric Orbits
. 175
9.2
Umbilic-Likeness of Pliicker Orbits
. 178
9.3
Unitary Orbits of the Pliicker Action
. 181
9.4
Umbilic-Likeness of Unitary Orbits
. 184
Contents
vii
9.5 The
Segre
Action and Its Symmetric Orbits
. 195
9.6
The Veronese Action and Its Symmetric Orbits
. 197
9.7
The Problem of Umbilic-Likeness of Veronese Orbits
. 201
9.8
Umbilic-Likeness of Veronese-Grassmann Orbits
. 205
9.9
Detailed Analysis of a Model Case
. 214
10
Geometric Descriptions in General
.219
10.1
Products of Umbilic-Like Orbits
. 219
10.2
General
Semiparallel Submanifolds
and Their Adapted Frame
Bundles
. 223
10.3
Warped Products and Immersed Fibre Bundles
. 227
10.4 Semiparallel
Submanifolds of Cylindrical or Toroidal
Segre
Type
. 229
10.4.1
The case of umbilic-like
Segre
orbits
. 230
10.4.2
The case of nonumbilic-like
Segre
orbits
. 235
11
Isometric
Semiparallel
Immersions of Riemannian Manifolds of
Conullity Two
.237
11.1 Semiparallel
Submanifolds with Plane Generators of
Codimension
2. 237
11.2
Some Particular Cases
. 241
11.3
Semiparallel Manifolds of Conullity Two in General
. 242
12
Some Generalizations
.249
12.1
ł-Semiparallel
Submanifolds
. 249
12.2
On 2-Semiparallel Submanifolds
. 252
12.3
2-Semiparallel Surfaces in Space Forms
. 253
12.4
Recurrent and
Pseudoparallel
Submanifolds
. 261
12.5
Submanifolds with Semiparallel Tensor Fields
. 263
12.6
Examples: The Surfaces
. 266
12.6.1
Я
-semiparallel
and
Я
-parallel
surfaces
. 267
12.6.2
^-parallel surfaces
. 270
12.6.3
R-
огкгс-рагаііеі
surfaces
. 271
12.6.4
Г
-semiparallel
surfaces
. 271
12.7
W/c-Semiparallel Hypersurfaces and Ryan's Problem
. 272
12.8
Extended Ryan's Problem for Normally Flat Submanifolds
. 279
12.9
fi-Semiparallel but Not Semiparallel Normally Flat Submanifolds
of Codimension
2. 282
References
.287
Index
.303 |
| adam_txt |
Contents
0
Introduction
. 1
1
Preliminaries
. 7
1.1
Real Spaces with Bilinear Metric
. 7
1.2
Moving Frames
. 8
1.3
(Pseudo-)Riemannian Manifolds
. 10
1.4
Standard Models of Space and Spacetime Forms
. 11
1.5
Symmetric (Pseudo-)Riemannian Manifolds
. 13
1.6
Semisymmetric (Pseudo-)Riemannian Manifolds
. 16
2
Submanifolds in Space Forms
. 23
2.1
A Submanifold and Its Adapted Frame Bundle
. 23
2.2
Higher-Order Fundamental Forms
. 26
2.3
Fundamental Identities
. 29
2.4
Osculating and Normal Subspaces of Higher Order
. 29
3
Parallel Submanifolds
. 33
3.1
Parallel and
ír-Parallel
Submanifolds
. 33
3.2
Examples:
Segre
and Pliicker Submanifolds
. 36
3.3
Example: Veronese Submanifold
. 40
3.4
Parallel Submanifolds and the Gauss Map
. 43
3.5
Parallel Submanifolds and Local Extrinsic Symmetry
. 44
3.6
Complete Parallel Irreducible Submanifolds as Standard Imbedded
Symmetric
Ќ-Ѕрасеѕ
. 46
4 Semiparallel
Submanifolds
. 51
4.1
The
Semiparallel
Condition and Its Special Cases
. 51
4.2
The
Semiparallel
Condition from the Algebraic Viewpoint
. 54
4.3
Decomposition of
Semiparallel
Fundamental Triplets
. 57
4.4
Triplets of Large Principal Codimension
. 59
4.5 Semiparallel
Submanifolds as Second-Order Envelopes of Parallel
Submanifolds
. 63
Contents
4.6
Second-Order Envelope of
Segre Submanifolds
. 66
4.7
A New Approach to Veronese Submanifolds
. 70
Normally Flat
Semiparallel
Submanifolds
. 73
5.1
Principal Curvature Vectors and the
Semiparallel
Condition
. 73
5.2
Normally Flat Parallel Submanifolds
. 75
5.3
Adapted Frame Bundle for a Second-Order Envelope
. 78
5.4
Second-Order Envelope as Warped Product
. 80
5.5 Semiparallel
Submanifolds of Principal Codimension
1. 84
5.6
Semiparallel Submanifolds of Principal Codimension
2
in
Euclidean Space
. 89
5.7
Normally Flat Semiparallel Submanifolds of Principal
Codimension
2
in Non-Euclidean Space Forms
. 93
Semiparallel Surfaces
. 97
6.1
Semiparallel Spacelike Surfaces
. 97
6.2
The Case of Regular Metrics
. 99
6.3
Veronese Surfaces
. 101
6.4
Second-Order Envelopes of Veronese Surfaces
. 106
6.5
The Case of a Singular Metric
. 108
6.5.1
The subcases where
spanţA,
В}
has singular metric
. 109
6.5.2
The subcases where span{A, B} has regular metric
. 112
6.6
Semiparallel Timelike Surfaces in
Lorentz
Spacetime
Forms
. 114
6.6.1
The principal case
. 115
6.6.2
The exceptional case
. 119
6.7
Spacelike 2-Parallel Surfaces
. 123
6.8
^-Parallel Surfaces as Semiparallel Surfaces
. 130
Semiparallel Three-Dimensional Submanifolds
. 135
7.1
Semiparallel Submanifolds M3 of Principal Codimension m\
< 2. 135
7.2 Nonminimal
Semiparallel M3 of Principal Codimension
т\=Ъ
. 138
7.3
Semiparallel M3 of Principal Codimension m\—A
. 147
7.4
Higher Principal Codimensions: Conclusions
. 154
Decomposition Theorems
. 157
8.
1 Decomposition of Semiparallel Submanifolds
. 157
8.2
Decomposition of Parallel Submanifolds
. 162
8.3
Decomposition of Normally Flat 2-Parallel Submanifolds
. 164
8.4
Structure of Submanifolds with Flat van
der Waerden-Bortolotti
Connection
. 168
Umbilic-Likeness of Main Symmetric Orbits
. 175
9.1
Two Kinds of Symmetric Orbits
. 175
9.2
Umbilic-Likeness of Pliicker Orbits
. 178
9.3
Unitary Orbits of the Pliicker Action
. 181
9.4
Umbilic-Likeness of Unitary Orbits
. 184
Contents
vii
9.5 The
Segre
Action and Its Symmetric Orbits
. 195
9.6
The Veronese Action and Its Symmetric Orbits
. 197
9.7
The Problem of Umbilic-Likeness of Veronese Orbits
. 201
9.8
Umbilic-Likeness of Veronese-Grassmann Orbits
. 205
9.9
Detailed Analysis of a Model Case
. 214
10
Geometric Descriptions in General
.219
10.1
Products of Umbilic-Like Orbits
. 219
10.2
General
Semiparallel Submanifolds
and Their Adapted Frame
Bundles
. 223
10.3
Warped Products and Immersed Fibre Bundles
. 227
10.4 Semiparallel
Submanifolds of Cylindrical or Toroidal
Segre
Type
. 229
10.4.1
The case of umbilic-like
Segre
orbits
. 230
10.4.2
The case of nonumbilic-like
Segre
orbits
. 235
11
Isometric
Semiparallel
Immersions of Riemannian Manifolds of
Conullity Two
.237
11.1 Semiparallel
Submanifolds with Plane Generators of
Codimension
2. 237
11.2
Some Particular Cases
. 241
11.3
Semiparallel Manifolds of Conullity Two in General
. 242
12
Some Generalizations
.249
12.1
ł-Semiparallel
Submanifolds
. 249
12.2
On 2-Semiparallel Submanifolds
. 252
12.3
2-Semiparallel Surfaces in Space Forms
. 253
12.4
Recurrent and
Pseudoparallel
Submanifolds
. 261
12.5
Submanifolds with Semiparallel Tensor Fields
. 263
12.6
Examples: The Surfaces
. 266
12.6.1
Я
-semiparallel
and
Я
-parallel
surfaces
. 267
12.6.2
^-parallel surfaces
. 270
12.6.3
R-
огкгс-рагаііеі
surfaces
. 271
12.6.4
Г
-semiparallel
surfaces
. 271
12.7
W/c-Semiparallel Hypersurfaces and Ryan's Problem
. 272
12.8
Extended Ryan's Problem for Normally Flat Submanifolds
. 279
12.9
fi-Semiparallel but Not Semiparallel Normally Flat Submanifolds
of Codimension
2. 282
References
.287
Index
.303 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Lumiste, Ülo 1929- |
| author_GND | (DE-588)119126745 |
| author_facet | Lumiste, Ülo 1929- |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)81452957 (DE-599)BVBBV022824128 |
| dewey-full | 516.36222 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.362 22 |
| dewey-search | 516.362 22 |
| dewey-sort | 3516.362 222 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022824128 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T18:41:04Z |
| indexdate | 2024-07-20T09:23:51Z |
| institution | BVB |
| isbn | 9780387499116 9780387499130 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016029435 |
| oclc_num | 81452957 |
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| physical | VII, 306 S. |
| publishDate | 2009 |
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| publisher | Springer |
| record_format | marc |
| spelling | Lumiste, Ülo 1929- Verfasser (DE-588)119126745 aut Semiparallel submanifolds in space forms Ülo Lumiste New York Springer 2009 VII, 306 S. txt rdacontent n rdamedia nc rdacarrier Submanifolds Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Untermannigfaltigkeit (DE-588)4128503-7 gnd rswk-swf Untermannigfaltigkeit (DE-588)4128503-7 s Differentialgeometrie (DE-588)4012248-7 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2869769&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016029435&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lumiste, Ülo 1929- Semiparallel submanifolds in space forms Submanifolds Differentialgeometrie (DE-588)4012248-7 gnd Untermannigfaltigkeit (DE-588)4128503-7 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4128503-7 |
| title | Semiparallel submanifolds in space forms |
| title_auth | Semiparallel submanifolds in space forms |
| title_exact_search | Semiparallel submanifolds in space forms |
| title_exact_search_txtP | Semiparallel submanifolds in space forms |
| title_full | Semiparallel submanifolds in space forms Ülo Lumiste |
| title_fullStr | Semiparallel submanifolds in space forms Ülo Lumiste |
| title_full_unstemmed | Semiparallel submanifolds in space forms Ülo Lumiste |
| title_short | Semiparallel submanifolds in space forms |
| title_sort | semiparallel submanifolds in space forms |
| topic | Submanifolds Differentialgeometrie (DE-588)4012248-7 gnd Untermannigfaltigkeit (DE-588)4128503-7 gnd |
| topic_facet | Submanifolds Differentialgeometrie Untermannigfaltigkeit |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2869769&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016029435&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT lumisteulo semiparallelsubmanifoldsinspaceforms |