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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.
Book 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. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition 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. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful.
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3 人中、2人の方が、｢このレビューが参考になった｣と投票しています。
形式:ハードカバー|Amazonが確認した購入
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.
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.
Amazon.com で最も参考になったカスタマーレビュー (beta)
Amazon.com:
6件のカスタマーレビュー
23 人中、21人の方が、｢このレビューが参考になった｣と投票しています。
Great book, but not perfect or ideal for every purpose
2006/4/29
形式:ハードカバー
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.
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.
28 人中、24人の方が、｢このレビューが参考になった｣と投票しています。
A good book if you have some background
2001/11/26
形式:ハードカバー
This book is a good introduction to manifolds and lie groups.
Still if you dont have any background ,this is not the book to start with.The first chapter is about the basics of manifolds:vector fields,lie brackts,flows on manifolds and more,
this chapter can help one alot as a second book on the subject.
The second chapter is about tensors, this introduction can be very hard for someone who didnt met the notion of tensors ,since the book tends to take a very general line with out down to earth examples.the 3ed chapter is about lie groups.It is avery good introduction ,to my point of view ,one of the best there is.
The 4th chapter is about integration on manifolds and is very good too.Chapters 5and6 are about De Rham cohomology theory and the hodge theorem.
If you have some knowledge on all the above subjects this book can serve as a very good overview on the subject.
Still if you dont have any background ,this is not the book to start with.The first chapter is about the basics of manifolds:vector fields,lie brackts,flows on manifolds and more,
this chapter can help one alot as a second book on the subject.
The second chapter is about tensors, this introduction can be very hard for someone who didnt met the notion of tensors ,since the book tends to take a very general line with out down to earth examples.the 3ed chapter is about lie groups.It is avery good introduction ,to my point of view ,one of the best there is.
The 4th chapter is about integration on manifolds and is very good too.Chapters 5and6 are about De Rham cohomology theory and the hodge theorem.
If you have some knowledge on all the above subjects this book can serve as a very good overview on the subject.
11 人中、8人の方が、｢このレビューが参考になった｣と投票しています。
Good, as long as you have enough background
2003/11/22
形式:ハードカバー
I read this book at the very beginning of my studying in differential geometry and was striked. The definitions and methods used in this book seemed totally incomprehensible to me. However, after some development in this field, I found that this book is very concise. It is a very good surey on differential geometry but not a good book to start with. Definitions are given from the most "down to bottom" one. It is a very good attitude, yet, if you do not have much background in differential geometry, this book may takes you several days in order to understand the concept of tensor and exterior algebra.
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