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Intended for a one year course, this volume serves 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. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research.
Important additions to this new edition include:
- A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise
- An increased number of coordinate calculations of connection and curvature
- General fomulas for curvature on Lie Groups and submersions
- Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger
- Several recent results about manifolds with positive curvature
List of contents
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.
Summary
Intended for a one year course, this volume serves 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. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research.
Important additions to this new edition include:
- A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise
- An increased number of coordinate calculations of connection and curvature
- General fomulas for curvature on Lie Groups and submersions
- Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger
- Several recent results about manifolds with positive curvature
Additional text
From the reviews of the second edition:P. Petersen
Riemannian 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)
Report
From the reviews of the second edition:P. Petersen
Riemannian 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)
