Differential geometry of curves and surfaces:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Englewood Cliffs, N.J.
Prentice-Hall
1976
|
| Ausgabe: | 1. print. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Portug. übers. - Hier auch unveränderte Nachdrucke desselben Jahres |
| Beschreibung: | VIII, 503 S. graph. Darst. |
| ISBN: | 0132125897 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804135849456041984 |
|---|---|
| adam_text | Contents
Preface v
Some Remarks on Using this Book vii
1. Curves 1
1 1 Introduction 1
1 2 Parametrized Curves 2
1 3 Regulär Curves; Are Length 5
1 4 The Vector Product in R3 11
1 5 The Local Theory of Curves Parametrized by Are Length 16
1 6 The Local Canonical Form 27
1 7 Global Properties of Plane Curves 30
2. Regulär Surfaces 51
2 1 Introduction 5/
2 2 Regulär Surfaces; Inverse Images of Regulär Values 52
2 3 Change of Parameters; Differential Functions on Surfaces 69
2 4 The Tangent Plane; the Differential of a Map 83
2 5 The First Fundamental Form; Area 92
2 6 Orientation of Surfaces 102
2 7 A Characterizauon of Compact Orientable Surfaces 109
2 8 A Geometrie Definition of Area 114
Appendix: A Brief Review on Continuity
and Differentiability 118
in
iv Contents
3. The Geometry of the Gauss Map 134
3 1 Introduction 134
3 2 The Definition of the Gauss Map and
Its Fundamental Properties 135
3 3 The Gauss Map in Local Coordinates 153
3 4 Vector Fields 175
3 5 Ruled Surfaces and Minimal Surfaces 188
Appendix: Self Adjoint Linear Maps and Quadratic Forms 214
4. The Intrinsic Geometry of Surfaces 217
4 1 Introduction 217
4 2 Isometries; Conformal Maps 218
4 3 The Gauss Theorem and the Equations of Compatibility 231
4 4 Parallel Transport; Geodesics 238
4 5 The Gauss Bonnet Theorem and its Applications 264
4 6 The Exponential Map. Geodesic Polar Coordinates 283
4 7 Further Properties of Geodesics. Convex Neighborhoods 298
Appendix: Proofs of the Fundamental Theorems of
The Local Theory of Curves and Surfaces 309
5. Global Differential Geometry 315
5 1 Introduction 315
5 2 The Rigidity of the Sphere 317
5 3 Complete Surfaces. Theorem of Hopf Rinow 325
5 4 First and Second Variations of the Are Length;
Bonnet s Theorem 339
5 5 Jacobi Fields and Conjugate Points 357
5 6 Covering Spaces; the Theorems of Hadamard 371
5 7 Global Theorems for Curves; the Fary Milnor Theorem 380
5 8 Surfaces of Zero Gaussian Curvature 408
5 9 Jacobi s Theorems 415
5 10 Abstract Surfaces; Further Generalizations 425
5 11 Hilbert s Theorem 446
Appendix: Point Set Topology of Euclidean Spaces 456
Bibliography and Comments 471
Hints and Answers to Some Exercises 475
Index 497
|
| adam_txt |
Contents
Preface v
Some Remarks on Using this Book vii
1. Curves 1
1 1 Introduction 1
1 2 Parametrized Curves 2
1 3 Regulär Curves; Are Length 5
1 4 The Vector Product in R3 11
1 5 The Local Theory of Curves Parametrized by Are Length 16
1 6 The Local Canonical Form 27
1 7 Global Properties of Plane Curves 30
2. Regulär Surfaces 51
2 1 Introduction 5/
2 2 Regulär Surfaces; Inverse Images of Regulär Values 52
2 3 Change of Parameters; Differential Functions on Surfaces 69
2 4 The Tangent Plane; the Differential of a Map 83
2 5 The First Fundamental Form; Area 92
2 6 Orientation of Surfaces 102
2 7 A Characterizauon of Compact Orientable Surfaces 109
2 8 A Geometrie Definition of Area 114
Appendix: A Brief Review on Continuity
and Differentiability 118
in
iv Contents
3. The Geometry of the Gauss Map 134
3 1 Introduction 134
3 2 The Definition of the Gauss Map and
Its Fundamental Properties 135
3 3 The Gauss Map in Local Coordinates 153
3 4 Vector Fields 175
3 5 Ruled Surfaces and Minimal Surfaces 188
Appendix: Self Adjoint Linear Maps and Quadratic Forms 214
4. The Intrinsic Geometry of Surfaces 217
4 1 Introduction 217
4 2 Isometries; Conformal Maps 218
4 3 The Gauss Theorem and the Equations of Compatibility 231
4 4 Parallel Transport; Geodesics 238
4 5 The Gauss Bonnet Theorem and its Applications 264
4 6 The Exponential Map. Geodesic Polar Coordinates 283
4 7 Further Properties of Geodesics. Convex Neighborhoods 298
Appendix: Proofs of the Fundamental Theorems of
The Local Theory of Curves and Surfaces 309
5. Global Differential Geometry 315
5 1 Introduction 315
5 2 The Rigidity of the Sphere 317
5 3 Complete Surfaces. Theorem of Hopf Rinow 325
5 4 First and Second Variations of the Are Length;
Bonnet's Theorem 339
5 5 Jacobi Fields and Conjugate Points 357
5 6 Covering Spaces; the Theorems of Hadamard 371
5 7 Global Theorems for Curves; the Fary Milnor Theorem 380
5 8 Surfaces of Zero Gaussian Curvature 408
5 9 Jacobi's Theorems 415
5 10 Abstract Surfaces; Further Generalizations 425
5 11 Hilbert's Theorem 446
Appendix: Point Set Topology of Euclidean Spaces 456
Bibliography and Comments 471
Hints and Answers to Some Exercises 475
Index 497 |
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| author | Carmo, Manfredo Perdigão do 1928-2018 |
| author_GND | (DE-588)1022011774 |
| author_facet | Carmo, Manfredo Perdigão do 1928-2018 |
| author_role | aut |
| author_sort | Carmo, Manfredo Perdigão do 1928-2018 |
| author_variant | m p d c mpd mpdc |
| building | Verbundindex |
| bvnumber | BV021902617 |
| callnumber-first | Q - Science |
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| callnumber-raw | QA641 C3713 1976 |
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| callnumber-sort | QA 3641 C3713 41976 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)299416343 (OCoLC)605257503 (DE-599)BVBBV021902617 |
| dewey-full | 516.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 |
| dewey-search | 516.36 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. print. |
| format | Book |
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| id | DE-604.BV021902617 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:04:45Z |
| indexdate | 2024-07-09T20:47:02Z |
| institution | BVB |
| isbn | 0132125897 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015117797 |
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| physical | VIII, 503 S. graph. Darst. |
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| publisher | Prentice-Hall |
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| spelling | Carmo, Manfredo Perdigão do 1928-2018 Verfasser (DE-588)1022011774 aut Differential geometry of curves and surfaces Manfredo P. do Carmo 1. print. Englewood Cliffs, N.J. Prentice-Hall 1976 VIII, 503 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aus dem Portug. übers. - Hier auch unveränderte Nachdrucke desselben Jahres Courbes Géométrie différentielle Géométrie différentielle ram Surfaces (Mathématiques) courbe inriac géométrie différentielle inriac surface inriac Flächentheorie (DE-588)4475211-8 gnd rswk-swf Fläche (DE-588)4129864-0 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Oberfläche (DE-588)4042907-6 gnd rswk-swf Kurve (DE-588)4033824-1 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Matériel didactique Differentialgeometrie (DE-588)4012248-7 s DE-604 Oberfläche (DE-588)4042907-6 s Kurve (DE-588)4033824-1 s 2\p DE-604 Fläche (DE-588)4129864-0 s 3\p DE-604 Flächentheorie (DE-588)4475211-8 s 4\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015117797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Carmo, Manfredo Perdigão do 1928-2018 Differential geometry of curves and surfaces Courbes Géométrie différentielle Géométrie différentielle ram Surfaces (Mathématiques) courbe inriac géométrie différentielle inriac surface inriac Flächentheorie (DE-588)4475211-8 gnd Fläche (DE-588)4129864-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd Oberfläche (DE-588)4042907-6 gnd Kurve (DE-588)4033824-1 gnd |
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| title | Differential geometry of curves and surfaces |
| title_auth | Differential geometry of curves and surfaces |
| title_exact_search | Differential geometry of curves and surfaces |
| title_exact_search_txtP | Differential geometry of curves and surfaces |
| title_full | Differential geometry of curves and surfaces Manfredo P. do Carmo |
| title_fullStr | Differential geometry of curves and surfaces Manfredo P. do Carmo |
| title_full_unstemmed | Differential geometry of curves and surfaces Manfredo P. do Carmo |
| title_short | Differential geometry of curves and surfaces |
| title_sort | differential geometry of curves and surfaces |
| topic | Courbes Géométrie différentielle Géométrie différentielle ram Surfaces (Mathématiques) courbe inriac géométrie différentielle inriac surface inriac Flächentheorie (DE-588)4475211-8 gnd Fläche (DE-588)4129864-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd Oberfläche (DE-588)4042907-6 gnd Kurve (DE-588)4033824-1 gnd |
| topic_facet | Courbes Géométrie différentielle Surfaces (Mathématiques) courbe géométrie différentielle surface Flächentheorie Fläche Differentialgeometrie Oberfläche Kurve Einführung Matériel didactique |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015117797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT carmomanfredoperdigaodo differentialgeometryofcurvesandsurfaces |