Riemannian Geometry (Graduate Texts in Mathematics Book 171) 2nd Edition by Peter Petersen (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a substantial graduate level book on differential geometry which assumes that you have already read a few of the standard introductory textbooks. This is definitely not a beginner’s book. Petersen’s ”

⭐” is a textbook which prepares the reader for research in this subject. It introduces the reader to many of the research topics, techniques and general way of thinking of Riemannian geometry related research. This book is closely related to pure mathematics research areas, not mathematical physics.Since this book is not a first introduction to Riemannian geometry, the reader thinking of acquiring this book will probably already have some familiarity with the topics and basic concepts listed in the table of contents. So it is not truly necessary to give a review. Anyone thinking of buying it will probably know people who have recommended it for specific purposes. At the graduate level in this subject, very few people are likely to rely on an Amazon review to make a decision to buy it or not.I’ll just repeat….. this is an advanced book which prepares the reader for research. Make sure you’ve read and understood at least a few modern differential geometry books first before acquiring it. Otherwise you’re going to spend a really long time staring at all those symbols, trying to work out what they mean!

⭐Dear SirsIt is only true that Bernard Riemann was one the greatest mathematicians that lived, his works were fundamental for mathematics, vector and mainly tensor calculus that would be the main base of relativity, perhaps he was a mathematician better then Eistein, but he didn’t have the fundamental creative genius of Einstein, he developed tensor calculus as well as Kronecker, Levi-Civita, Ricci, his works are there for of the most importance for understang relativity, calculus and mathematics, this book as donated by me to the Lisbon “Tecnico”, is a clear book in the purpose of enlightening Riemann mathematics.P. Rose/ M.Lapa

⭐While this book is a fantastic read and an invaluable reference, it was delivered in a terrible condition with scratches and dirt on the cover. If I had wanted a second hand-looking book, I would have ordered one on purpose.

⭐Advanced treatise on Riemannian Geometry, and a standard reference for advanced graduate students. Prerequisites include knowledge of smooth manifold theory at the level of Lee’s tome.

⭐É um produto de excelente qualidade.

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⭐1960年代後半から今日に至る40年余りの間に、リーマン幾何学のComparison Geometryと呼ばれる分野は非常に大きく進展した。本書はチーガーの有限性定理とグロモフの収束定理が大きな推進力であったこの分野を解説するテキストレベルの初の成書であり、待望の一冊といえる。本書を一読すれば、リーマン多様体上の距離関数に関する勾配場、ヘッシャン、ラプラシアンが全体を通して非常に重要な役割を果たしている事が読み取れる。特に、距離関数のヘッシャンが曲率を関数項に含んだリッカチ方程式（本書で基本方程式と呼ぶ）の解になるという事実が重要で、これからリーマン幾何の比較定理の多くが二つのリッカチ型方程式の解の比較に帰着するという事情も理解できる（興味ある方は、加須栄先生の『リーマン幾何学』の第4章と比較・検討してみて頂きたい）。基本方程式を前面に出し、ヤコビ場を用いる場合と同等の結論が導ける事を明示している点に、他の成書にない本書の最大の特徴がある。本書ではボホナー技法も詳しく解説されており、キリングベクトル場とガウス・ボンネの定理に関する基本予想との関連や非負の曲率作用素をもつコンパクトリーマン多様体を分類するギャロ・マイヤーの素敵な定理が述べられている。本書最大のハイライトはComparison Geometryの主要成果を述べた第9〜第11章にある。ここではグロモフやチーガーなどのスーパースターの素晴らしいアイディアと成果が解説されていてとても面白い。一方、最後の二つの章は本書で最も難しい所であるので、じっくりと取り組む必要がある。本書の説明では良く分からないと感じる時に、酒井先生とChavelのリーマン幾何のテキストが良き相談相手になってくれる場合もあると思う。本書の読者の方に、Bergerのパノラマ本（”A Panoramic View of Riemannian Geometry”）の第6,7,11,12,13章などを参照されると更に広汎な展望が得られるので、併読されると面白いと思う。

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⭐Great book, but clearly not for readers who want an introduction to the subject.

⭐This book requires too much prerequisites without guiding a novice, from where he could find those prerequisites easily with the same notations he used. The author discusses took many topics scantly, rather than a few topics with enough detail so that definite progress on the part of reader’s side is ensured. However, the book looks gregarious.

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