- Series: Universitext
- Paperback: 611 pages
- Publisher: Springer; 6th ed. 2011 edition (August 9, 2011)
- Language: English
- ISBN-10: 3642212972
- ISBN-13: 978-3642212970
- Product Dimensions: 6 x 1.2 x 9 inches
- Shipping Weight: 2 pounds
- Average Customer Review: 1 customer review
- Amazon Best Sellers Rank: #2,636,839 in Books (See Top 100 in Books)
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Riemannian Geometry and Geometric Analysis (Universitext) 6th ed. 2011 Edition
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From the reviews of the sixth edition:
“For readers familiar with the basics of differential geometry and some acquaintance with modern analysis, the book is reasonably self-contained … . The book succeeds very well in laying out the foundations of modern Riemannian geometry and geometric analysis. It introduces a number of key techniques and provides a representative overview of the field. This sixth edition contains a deeper study of the spectrum of the Laplace operator and its relation to the geometry of the underlying Riemannian manifold.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 171 (3-4), September, 2013)
“The present book is the sixth edition of the author’s textbook on Riemannian geometry and geometric analysis. In general, it gives a synthesis of geometric and analytic methods in the study of Riemannian manifolds. … the book is well written. That is why I would recommend it to specialists in mathematics or physics and especially to Ph.D. students of differential geometry.” (Miroslaw Doupovec, Zentralblatt MATH, Vol. 1227, 2012)
From the Back Cover
This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. The previous edition already introduced and explained the ideas of the parabolic methods that had found a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discussed further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry.
The 6th edition includes a systematic treatment of eigenvalues of Riemannian manifolds and several other additions. Also, the entire material has been reorganized in order to improve the coherence of the book.
From the reviews:
"This book provides a very readable introduction to Riemannian geometry and geometric analysis. ... With the vast development of the mathematical subject of geometric analysis, the present textbook is most welcome." Mathematical Reviews
"...the material ... is self-contained. Each chapter ends with a set of exercises. Most of the paragraphs have a section ‘Perspectives’, written with the aim to place the material in a broader context and explain further results and directions." Zentralblatt MATH
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Overall, I would recommend this book for its modern approach, though it was there in Kobayashi-Nomizu but is unfortunately not much in use, however, I must say that it must be supplemented by Petersen for topics like comparison geometry and material in later chapters of Petersen which is some what unique feature of Petersen's book. I havent read Chavel but I guess this book under review will come close to Chavel's than rest of books mentioned above.
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