Riemannian Geometry
(Sprache: Englisch)
This comprehensive introduction to Riemannian Geometry offers a detailed and engaging account of the topic, plus numerous exercises and examples. It combines both the geometric parts of Riemannian geometry and the analytic aspects of the theory, and reviews the latest research.
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Produktinformationen zu „Riemannian Geometry “
This comprehensive introduction to Riemannian Geometry offers a detailed and engaging account of the topic, plus numerous exercises and examples. It combines both the geometric parts of Riemannian geometry and the analytic aspects of the theory, and reviews the latest research.
Klappentext zu „Riemannian Geometry “
This volume introduces techniques and theorems of Riemannian geometry, and opens the way to advanced topics. The text combines the geometric parts of Riemannian geometry with analytic aspects of the theory, and reviews recent research. The updated second edition includes a new coordinate-free formula that is easily remembered (the Koszul formula in disguise); an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus integrated into the text, allowing for an early treatment of the Sphere theorem using a forgotten proof by Berger; recent results regarding manifolds with positive curvature.
This book is intended for a one year course in Riemannian Geometry. It will serve as a single source, introducing students to the important techniques and theorems while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian Geometry. Instead of variational techniques, the author uses a unique approach emphasizing distance functions and special coordinate systems. He also uses standard calculus with some techniques from differential equations, instead of variational calculus, thereby providing a more elementary route for students. Many of the chapters contain material typically found in specialized texts and never before published together in one source. Key sections include noteworthy coverage of: geodesic geometry, Bochner technique, symmetric spaces, holonomy, comparison theory for both Ricci and sectional curvature, and convergence theory. This volume is one of the few published works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory as well as presenting the most up-to-date research including sections on convergence and compactness of families of manifolds. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stoke's theorem. Scattered throughout the text is a variety of exercises which will help to motivate readers to deepen their understanding of the subject.
Inhaltsverzeichnis zu „Riemannian Geometry “
- Introduction- Riemannian Metrics
- Curvature
- Examples
- Hypersurfaces
- Geodesics and Distance
- Sectional Curvature Comparison I
- The Bochner Technique
- Symmetric Spaces and Holonomy
- Ricci Curvature Comparison
- Convergence
- Sectional Curvature Comparison II
- Appendix A: De Rham Cohomology
- Bibliography
- Index
Bibliographische Angaben
- Autor: Peter Petersen
- 2006, 2nd ed., 405 Seiten, 59 Schwarz-Weiß-Abbildungen, Maße: 15,5 x 23,5 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 0387292462
- ISBN-13: 9780387292465
- Erscheinungsdatum: 09.08.2006
Sprache:
Englisch
Rezension zu „Riemannian Geometry “
From the reviews of the second edition:P. PetersenRiemannian Geometry"A nice introduction to Riemannian geometry, containing basic theory as well as several advanced topics."-EUROPEAN MATHEMATICAL SOCIETY "This is an introduction to modern methods in Riemannian geometry containing interesting and original approaches to many areas in this field. ... After a general introduction (metrics, curvature, geodesics) and concrete calculations for many examples, the second half of the book considers Bochner-Cartan techniques and comparison geometry. Particularly for these aspects it continues to play an outstanding role among textbooks in Riemannian geometry." (M. Kunzinger, Monatshefte für Mathematik, Vol. 154 (1), May, 2008)
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