《微分流形導論（第2版）》是2010年世界圖書出版公司出版的圖書，作者是朗。
基本介紹
- 中文名：微分流形導論（第2版）
- 作者：朗
- ISBN：9787510027468
- 出版社：世界圖書出版公司
- 出版時間：2010-09-01
內容簡介,作品目錄,
內容簡介
this book is an outgrowth of my introduction to differentiable manifolds （1962） and differential manifolds （1972）. both i and my publishers felt it worth while to keep available a brief introduction to differential manifolds.
the book gives an introduction to the basic concepts which are used in differential topology， differential geometry， and differential equations. in differential topology， one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them （immersions， embeddings， isomorphisms， etc.）. one may also use differentiable structures on topological manifolds to determine the topological structure of the manifold （for example， a la smale [sm 67]）. in differential geometry， one puts an additional structure on the differentiable manifold （a vector field， a spray， a 2-form， a riemannian metric， ad lib.） and studies properties connected especially with these objects. formally， one may say that one studies properties invariant under the group of. differentiable automorphisms which preserve the additional structure. in differential equations， one studies vector fields and their integral curves， singular points， stable and unstable manifolds， etc. a certain number of concepts are essential for all three， and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings. the concepts are concerned with the general basic theory of differential manifolds. my fundamentals of differential geometry （1999） can then be viewed as a continuation of the present book.
作品目錄
Acknowledgments
CHAPTER I
Differential Calculus
1. Categories
2. Finite Dimensional Vector Spaces
3. Derivatives and Composition of Maps
4. Integration and Tayiors Formula
5. The Inverse Mapping Theorem
CHAPTER II
Manifolds
1. Atlases， Charts， Morphisms
2. Submanifolds， Immersions， Submersions
3. Partitions of Unity
4. Manifolds with Boundary
CHAPTER III
Vector Bundles
l. Definition， Pull Backs
2. The Tangent Bundle
3. Exact Sequences of Bundles
4. Operations on Vector Bundles
5. Splitting of Vector Bundles
CHAPTER IV
Vector Fields and Differential Equations
1. Existence Theorem for Differential Equations
2. Vector Fields， Curves， and Flows
3. Sprays
4. The Flow of a Spray and the Exponential Map
5. Existence of Tubular Neighborhoods
6. Uniqueness of Tubular Neighborhoods
CHAPTER V
Oiretions on Vector Fields end Differential Forms
1. Vector Fields， Differential Operators， Brackets
2. Lie Derivative
3. Exterior Derivative
4. The Poincare Lemma
5. Contractions and Lie Derivative
6. Vector Fields and l-Forms Under Self Duality
7. The Canonical 2-Form
8. Darbouxs Theorem
CHAPTER VI
The Theorem of Frobenius
1. Statement of the Theorem
2. Differential Equations Depending on a Parameter
3. Proof of the Theorem
4. The Global Formulation
5. Lie Groups and Subgroups
CHAPTER VII
Metrics
1. Definition and Functoriality
2. The Metric Group
3. Reduction to the Metric Group
4. Metric Tubular Neighborhoods
5. The Morse Lemma
6. The Riemannian Distance
7. The Canonical Spray
CHAPTER VIII
Integretion of Differential Forms
1. Sets of Measure 0
2. Change of Variables Formula
3. Orientation
4. The Measure Associated with a Differential Form
CHAPTER IX
Stokes Theorem
1. Stokes Theorem for a Rectangular Simplex
2. Stokes Theorem on a Manifold
3. Stokes Theorem with Singularities
CHAPTER X
Applications of Stokes Theorem
1. The Maximal de Rham Cohomology
2. Volume forms and the Divergence
3. The Divergence Theorem
4. Cauchys Theorem
5. The Residue Theorem
Bibliography
Index
