Riemannian Manifolds: An Introduction to Curvature
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1997
|
| Series: | Graduate Texts in Mathematics
176 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | Thisbookisdesignedasatextbookforaone-quarterorone-semestergr- uate course on Riemannian geometry, for students who are familiar with topological and di?erentiable manifolds. It focuses on developing an in- mate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. I have selected a set of topics that can reasonably be covered in ten to ?fteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machineryofmetrics,connections,andgeodesics,withoutwhichonecannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all e?orts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing thetotalcurvatureofasurfaceintermsofitstopologicaltype),theCartan– Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose– Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints |
| Physical Description: | 1 Online-Ressource (XV, 226 p) |
| ISBN: | 9780387227269 9780387983226 |
| ISSN: | 0072-5285 |
| DOI: | 10.1007/b98852 |
Staff View
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| dewey-hundreds | 500 - Natural sciences and mathematics |
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| isbn | 9780387227269 9780387983226 |
| issn | 0072-5285 |
| language | English |
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| spelling | Lee, John M. Verfasser aut Riemannian Manifolds An Introduction to Curvature by John M. Lee New York, NY Springer New York 1997 1 Online-Ressource (XV, 226 p) txt rdacontent c rdamedia cr rdacarrier Graduate Texts in Mathematics 176 0072-5285 Thisbookisdesignedasatextbookforaone-quarterorone-semestergr- uate course on Riemannian geometry, for students who are familiar with topological and di?erentiable manifolds. It focuses on developing an in- mate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. I have selected a set of topics that can reasonably be covered in ten to ?fteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machineryofmetrics,connections,andgeodesics,withoutwhichonecannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all e?orts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing thetotalcurvatureofasurfaceintermsofitstopologicaltype),theCartan– Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose– Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints Mathematics Global differential geometry Differential Geometry Mathematik Krümmung (DE-588)4128765-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 s Krümmung (DE-588)4128765-4 s 1\p DE-604 Riemannscher Raum (DE-588)4128295-4 s 2\p DE-604 https://doi.org/10.1007/b98852 Verlag Volltext 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 |
| spellingShingle | Lee, John M. Riemannian Manifolds An Introduction to Curvature Mathematics Global differential geometry Differential Geometry Mathematik Krümmung (DE-588)4128765-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4128765-4 (DE-588)4128295-4 (DE-588)4128462-8 |
| title | Riemannian Manifolds An Introduction to Curvature |
| title_auth | Riemannian Manifolds An Introduction to Curvature |
| title_exact_search | Riemannian Manifolds An Introduction to Curvature |
| title_full | Riemannian Manifolds An Introduction to Curvature by John M. Lee |
| title_fullStr | Riemannian Manifolds An Introduction to Curvature by John M. Lee |
| title_full_unstemmed | Riemannian Manifolds An Introduction to Curvature by John M. Lee |
| title_short | Riemannian Manifolds |
| title_sort | riemannian manifolds an introduction to curvature |
| title_sub | An Introduction to Curvature |
| topic | Mathematics Global differential geometry Differential Geometry Mathematik Krümmung (DE-588)4128765-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Mathematics Global differential geometry Differential Geometry Mathematik Krümmung Riemannscher Raum Riemannsche Geometrie |
| url | https://doi.org/10.1007/b98852 |
| work_keys_str_mv | AT leejohnm riemannianmanifoldsanintroductiontocurvature |