Peter Petersen

Riemannian Geometry (eBook, PDF)

• Format: PDF

Produktbeschreibung

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. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.

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.

From reviews of the first edition:

"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."

- Bernd Wegner, Zentralblatt

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 405
• Erscheinungstermin: 24. November 2006
• Englisch
• ISBN-13: 9780387294032
• Artikelnr.: 43729226

Autorenporträt

Peter Petersen is a Professor of Mathematics at UCLA. His current research is on various aspects of Riemannian geometry. Professor Petersen has authored two important textbooks for Springer: Riemannian Geometry in the GTM series and Linear Algebra in the UTM series.

Inhaltsangabe

Preface.- 1. Riemannian Metrics.-2. Derivatives.- 3. Curvature.- 4. Examples.- 5. Geodesics and Distance.- 6. Sectional Curvature Comparison I.- 7. Ricci Curvature Comparison.- 8. Killing Fields.- 9. The Bochner Technique.- 10. Symmetric Spaces and Holonomy.- 11. Convergence.- 12. Sectional Curvature Comparison II.- Bibliography.- Index.

Rezensionen

"This is a very advanced textbook on metric and algebraic proofs of critical theorems in the field of metric spaces involving manifolds and other 3D structures. ... First, definitions, theorems, proofs, and exercises abound throughout every section of this 500 page mathematics book. The history of development in the area is comprehensive. ... The experts will find this a useful research tool. ... I recommend this book for researchers having a strong background to begin with." (Joseph J. Grenier, Amazon.com, June, 2016)

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)

P. Petersen Riemannian Geometry "A nice introduction to Riemannian geometry, containing basic theory as well as several advanced topics." -EUROPEAN MATHEMATICAL SOCIETY