Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts in Mathematics, 94) by Frank W. Warner (PDF)

Description

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

User’s Reviews

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

⭐This is an incredible textbook. Great for graduate students who have taken a course in vector calculus of differentiable topology. Not heavy on structure theorems or representation theory but fantastic for basics of Lie groups.

⭐It’s a classic book. A reference in Lie groups.

⭐If you are looking for a great reference for differentiable manifolds this is it! Be careful though it is dense at times.

⭐This is my first and my last kindle edition purchase. The book was traslated without any care about mathematical notations. The paper edition is fine, kindle edition is very very bad.

⭐Very well job done.

⭐This is a solid introduction to the foundations (and not just the basics) of differential geometry. The author is rather laconic, and the book requires one to work through it, rather than read it. It presupposes firm grasp of point-set topology, including paracompactness and normality. The basics (Inverse and Implicit Function Theorems, Frobenius Theorem, orientation, and rudiments of de Rham cohomology) are covered in about 100 pages (Chapters 1, 2, and 4). This is not really suitable for an undergraduate course in differential geometry, but is great for a graduate course.Chapter 3, 5, and 6 (self-contained introductions to Lie Groups, Sheaf Theory, and Hodge Theory, all from a geometric viewpoint) are a really nice feature. The book can’t be covered in one semester, but these chapters are great for selft-study. In fact, the organization of Chapter 5 is more suitable for self-study than for being taught in class (lots of theory developed first, with all applications delayed until the end). The real jewel of the book is Chapter 6, a very clean introduction to Hodge Theory, with immediate applications.The main drawback of the book in my view is that the author avoids vector bundles like the plague. These could have been very nicely incorporated into the book. No mention is made of Mayer-Vietoris or Kunneth formula, even though the former follows easily from the section on cochain complexes in Chapter 5 and the latter with some effort from Chapter 6. There is no mention of manifolds with boundary either, except as regular domains of manifolds for the purpose of Stokes Theorem.The organization of the book could have been better as well. In particular, the section on cochain complexes could have been incorporated in the rather short de Rham Cohomology Chapter 4, so that MV could have been proved and used to compute the cohomology of spheres (beyond the circle). Some subsections, including in Chapter 1, appear out of order to me. There is a shortage of exercises in my view. Some of the author’s notation (for tangent spaces, tangent bundles) is rather non-standard.However, all-in-all, I can’t think of a better differential geometry text for a graduate course. Spivak and Lee are quite wordy and do not have the same breadth. Either book would be preferable to Warner for an undergraduate course though. The price is a relative bargain too.

⭐If you’re a math professional looking for a great Differential Geometry reference with details to many proofs, this book is for you.If you are a student and you need to use this for a graduate course, Good luck. There are little to no examples of any of the notions or notations used in the book.To make things concrete, you will have to make up examples in R^n on your own, or hope your professor provides information to clarify.

⭐This book is superb at the graduate level. If you are an ambitious undergraduate, start with “Differential Forms, a Complement to Vector Calculus” by Steven H. Weintraub and “Calculus on Manifolds” by Michael Spivak. Proceed to “Lecture Notes on Elementary Topology and Geometry” by Singer and Thorpe. Then read this book. It’s the real deal!

⭐great book, but print quality is not optimal.

⭐本書の大きなテーマは, 二つある.まず, 第１, 2章では, 多様体とベクトル場の基本を述べる. 特に, 第2章では, 積分多様体に関するフロベニウスの定理が証明され, 更にそれは differential ideal の概念と関連付けられ, 大域的な考察が成されている. その上で,一つ目のテーマとしては, リー群に関するものを述べている. 「リー群の閉部分群は, 再びリー部分群になる.」及び, 「リー群の閉部分群による商には、多様体構造が付く.」 と言う二つの定理が, 第3章で述べられる. ちなみに, リー群の閉部分群による商に関しては, Bourbaki 「Elements de mathematique」の多様体の巻では更に一般化され, 一般の可微分多様体上の同値関係による商構造がどのような場合に与えられるかと言う定理が述べられており, 主バンドルの理論への応用がある.二つ目は、de Rham の定理である.第 4, 5章において, 「de Rham cohomology と singular cohomology の graded algebra としての標準的な同型」が定義され, それが実際に同型であることが証明される. graded algebra としての同型定理ではなく, graded group としての同型定理は, 「シンガー＆ソープ：トポロジーと幾何学入門」と言う本にも書いてあるが, シンガーとソープの本での証明は, 多様体の三角形分割の定理を暗に必要としている.しかし, この Warner の本での証明は, 層の理論を用いることにより, 三角形分割定理を使用することなく, graded algebra としての同型定理の証明を, 綺麗に述べている.最後に, 第6章では, 「コンパクトで向き付け可能なリーマン多様体の de Rham コホモロジー群の元は, ただ一つの調和形式で代表される」 という, Hodge の定理が証明されている. ただ, その証明に, 超関数の議論が用いられていることは, 意外であった.

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