Minimal submanifolds in Pseudo-Riemannian geometry:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2011
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 161 - 164 |
| Physical Description: | XV, 167 S. graph. Darst. |
| ISBN: | 9814291242 9789814291248 |
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Record in the Search Index
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| adam_text | Titel: Minimal submanifolds in pseudo-Riemannian geometry
Autor: Anciaux, Henri
Jahr: 2011
Contents
Foreword vii
Preface ix
1. Submanifolds in pseudo-Riemannian geometry 1
1.1 Pseudo-Riemannian manifolds................ 1
1.1.1 Pseudo-Riemannian metrics............. 1
1.1.2 Structures induced by the metric.......... 3
1.1.3 Calculus on a pseudo-Riemannian manifold .... 8
1.2 Submanifolds......................... 9
1.2.1 The tangent and the normal spaces......... 9
1.2.2 Intrinsic and extrinsic structures of a submanifold 11
1.2.3 One-dimensional submanifolds: Curves....... 14
1.2.4 Submanifolds of co-dimension one: Hypersurfaces . 17
1.3 The variation formulae for the volume........... 18
1.3.1 Variation of a submanifold.............. 18
1.3.2 The first variation formula.............. 19
1.3.3 The second variation formula............ 23
1.4 Exercises............................ 27
2. Minimal surfaces in pseudo-Euclidean space 29
2.1 Intrinsic geometry of surfaces................ 29
2.2 Graphs in Minkowski space ................. 32
2.3 The classification of ruled, minimal surfaces........ 40
2.4 Weierstrass representation for minimal surfaces...... 47
2.4.1 The definite case................... 48
2.4.2 The indefinite case.................. 52
2.4.3 A remark on the regularity of minimal surfaces . . 54
2.5 Exercises............................ 54
xiv Minimal submanifolds in pseudo-Riemannian geometry
3. Equivariant minimal hypersurfaces in space forms 57
3.1 The pseudo-Riemannian space forms............ 57
3.2 Equivariant minimal hypersurfaces in pseudo-Euclidean
space.............................. 61
3.2.1 Equivariant hypersurfaces in pseudo-Euclidean
space.......................... 61
3.2.2 The minimal equation................ 63
3.2.3 The definite case (e,e ) = (1,1)........... 65
3.2.4 The indefinite positive case (e,e ) = (—1,1) .... 66
3.2.5 The indefinite negative case (e,e ) = (—1,-1) ... 67
3.2.6 Conclusion ...................... 68
3.3 Equivariant minimal hypersurfaces in pseudo-space forms 69
3.3.1 Totally umbilic hypersurfaces in pseudo-space
forms ......................... 69
3.3.2 Equivariant hypersurfaces in pseudo-space forms . 73
3.3.3 Totally geodesic and isoparametric solutions .... 75
3.3.4 The spherical case (e,e ,e ) = (1,1,1) ....... 76
3.3.5 The elliptic hyperbolic case
(e,e ,e ) = (1,-1,-1)................ 78
3.3.6 The hyperbolic hyperbolic case
(e,e ,e ) = (-1,-1,1)................ 80
3.3.7 The elliptic de Sitter case (e, e , e ) = (-1,1,1) . 81
3.3.8 The hyperbolic de Sitter case (e, e , e ) = (1, -1,1) 82
3.3.9 Conclusion ...................... 84
3.4 Exercises............................ 86
4. Pseudo-Kähler manifolds 89
4.1 The complex pseudo-Euclidean space............ 89
4.2 The general definition .................... 91
4.3 Complex space forms..................... 95
4.3.1 The case of dimension ? = 1............. 99
4.4 The tangent bundle of a pseudo-Kähler manifold ..... 100
4.4.1 The canonical symplectic structure of the
cotangent bundle T*M ............... 100
4.4.2 An almost complex structure on the tangent
bundle TM of a manifold equipped with an
affine connection................... 102
4.4.3 Identifying T*M and TM and the Sasaki metric . 104
Contents xv
4.4.4 A complex structure on the tangent bundle of a
pseudo-Kähler manifold............... 106
4.4.5 Examples....................... 108
4.5 Exercises............................ 109
5. Complex and Lagrangian submanifolds in pseudo-Kähler
manifolds 111
5.1 Complex submanifolds.................... Ill
5.2 Lagrangian submanifolds................... 113
5.3 Minimal Lagrangian surfaces in C2 with neutral metric . . 114
5.4 Minimal Lagrangian submanifolds in Cn.......... 116
5.4.1 Lagrangian graphs.................. 118
5.4.2 Equivariant Lagrangian submanifolds........ 120
5.4.3 Lagrangian submanifolds from evolving quadrics . 123
5.5 Minimal Lagrangian submanifols in complex space forms . 127
5.5.1 Lagrangian and Legendrian submanifolds ..... 128
5.5.2 Equivariant Legendrian submanifolds in
odd-dimensional space forms ............ 133
5.5.3 Minimal equivariant Lagrangian submanifolds in
complex space forms................. 137
5.6 Minimal Lagrangian surfaces in the tangent bundle of a
Riemannian surface...................... 143
5.6.1 Rank one Lagrangian surfaces............ 144
5.6.2 Rank two Lagrangian surfaces............ 146
5.7 Exercises............................ 148
6. Minimizing properties of minimal submanifolds 151
6.1 Minimizing submanifolds and calibrations......... 151
6.1.1 Hypersurfaces in pseudo-Euclidean space...... 151
6.1.2 Complex submanifolds in pseudo-Kähler manifolds 155
6.1.3 Minimal Lagrangian submanifolds in complex
pseudo-Euclidean space............... 156
6.2 Non-minimizing submanifolds................ 158
Bibliography 161
Index 165
|
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| ctrlnum | (OCoLC)730022278 (DE-599)GBV638789990 |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T23:23:19Z |
| institution | BVB |
| isbn | 9814291242 9789814291248 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022544887 |
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| owner_facet | DE-19 DE-BY-UBM DE-83 |
| physical | XV, 167 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Anciaux, Henri Verfasser (DE-588)14333381X aut Minimal submanifolds in Pseudo-Riemannian geometry Henri Anciaux Singapore [u.a.] World Scientific 2011 XV, 167 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 161 - 164 Pseudo-Riemannscher Raum (DE-588)4176163-7 gnd rswk-swf Minimale Untermannigfaltigkeit (DE-588)4338425-0 gnd rswk-swf Pseudo-Riemannscher Raum (DE-588)4176163-7 s Minimale Untermannigfaltigkeit (DE-588)4338425-0 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022544887&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Anciaux, Henri Minimal submanifolds in Pseudo-Riemannian geometry Pseudo-Riemannscher Raum (DE-588)4176163-7 gnd Minimale Untermannigfaltigkeit (DE-588)4338425-0 gnd |
| subject_GND | (DE-588)4176163-7 (DE-588)4338425-0 |
| title | Minimal submanifolds in Pseudo-Riemannian geometry |
| title_auth | Minimal submanifolds in Pseudo-Riemannian geometry |
| title_exact_search | Minimal submanifolds in Pseudo-Riemannian geometry |
| title_full | Minimal submanifolds in Pseudo-Riemannian geometry Henri Anciaux |
| title_fullStr | Minimal submanifolds in Pseudo-Riemannian geometry Henri Anciaux |
| title_full_unstemmed | Minimal submanifolds in Pseudo-Riemannian geometry Henri Anciaux |
| title_short | Minimal submanifolds in Pseudo-Riemannian geometry |
| title_sort | minimal submanifolds in pseudo riemannian geometry |
| topic | Pseudo-Riemannscher Raum (DE-588)4176163-7 gnd Minimale Untermannigfaltigkeit (DE-588)4338425-0 gnd |
| topic_facet | Pseudo-Riemannscher Raum Minimale Untermannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022544887&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT anciauxhenri minimalsubmanifoldsinpseudoriemanniangeometry |