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El. knyga: Introduction to Differential Manifolds

  • Formatas: PDF+DRM
  • Išleidimo metai: 29-Jul-2015
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319207353
Kitos knygos pagal šią temą:
Introduction to Differential ManifoldsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 29-Jul-2015
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319207353
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This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces.Its ambition is to give solid foundations. In particular, the introduction of "abstract" notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them.The book should be of interest t

o various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory.The original French text Introduction aux variétés différentielles has been a best-seller in its category in France for many years.Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs.

Differential Calculus.- Manifolds: The Basics.- From Local to Global.- Lie Groups.- Differential Forms.- Integration and Applications.- Cohomology and Degree Theory.- Euler-Poincaré and Gauss-Bonnet.

Recenzijos

The book gives a detailed introduction to the world of differentiable manifolds and is of possible interested to everybody who wants to acquire a basic knowledge of differential geometry. Each chapter concludes with a list of exercises, solutions are given in the appendix. (Volker Branding, zbMATH 1338.58001, 2016)

Chapter 1 Differential Calculus
1(48)
1.1 Introduction
1(3)
1.1.1 What Is Differential Calculus?
1(2)
1.1.2 In This
Chapter
3(1)
1.2 Differentials
4(7)
1.2.1 Definition and Basic Properties
4(3)
1.2.2 Three Fundamental Examples
7(3)
1.2.3 Functions of Class Cp
10(1)
1.3 The Chain Rule
11(3)
1.4 Local Invertibility
14(7)
1.4.1 Diffeomorphisms
14(2)
1.4.2 Local Diffeomorphisms
16(2)
1.4.3 Immersions, Submersions
18(3)
1.5 Submanifolds
21(8)
1.5.1 Basic Properties
21(2)
1.5.2 Examples: Spheres, Tori, and the Orthogonal Group
23(2)
1.5.3 Parametrizations
25(1)
1.5.4 Tangent Vectors, Tangent Space
26(3)
1.6 One-Parameter Subgroups of the Linear Group
29(4)
1.7 Critical Points
33(3)
1.8 Critical Values
36(2)
1.9 Differential Calculus in Infinite Dimensions
38(2)
1.10 Comments
40(2)
1.11 Exercises
42(7)
Chapter 2 Manifolds: The Basics
49(48)
2.1 Introduction
49(2)
2.1.1 A Typical Example: The Set of Lines in the Plane
49(2)
2.1.2 In This
Chapter
51(1)
2.2 Charts, Atlases
51(5)
2.2.1 From Topological to Smooth Manifolds
51(3)
2.2.2 First Examples
54(2)
2.3 Differentiable Functions: Diffeomorphisms
56(4)
2.4 Fundamental Theorem of Algebra
60(1)
2.5 Projective Spaces
61(6)
2.6 The Tangent Space: Maps
67(6)
2.6.1 Tangent Space. Linear Tangent Map
67(2)
2.6.2 Local Diffeomorphisms, Immersions, Submersions, Submanifolds
69(4)
2.7 Covering Spaces
73(10)
2.7.1 Quotient of a Manifold by a Group
74(7)
2.7.2 Simply Connected Spaces
81(2)
2.8 Countability at Infinity
83(2)
2.9 Comments
85(2)
2.10 Exercises
87(10)
Chapter 3 From Local to Global
97(50)
3.1 Introduction
97(1)
3.2 Bump Functions; Embedding Manifolds
98(5)
3.3 Derivations
103(7)
3.3.1 Derivation at a Point
103(3)
3.3.2 Another Point of View on the Tangent Space
106(2)
3.3.3 Global Derivations
108(2)
3.4 Image of a Vector Field: Bracket
110(3)
3.5 The Tangent Bundle
113(6)
3.5.1 The Manifold of Tangent Vectors
113(1)
3.5.2 Vector Bundles
114(3)
3.5.3 Vector Fields on Manifolds: The Hessian
117(2)
3.6 The Flow of a Vector Field
119(8)
3.7 Time-Dependent Vector Fields
127(4)
3.8 One-Dimensional Manifolds
131(2)
3.9 Comments
133(4)
3.10 Exercises
137(10)
Chapter 4 Lie Groups
147(38)
4.1 Introduction
147(1)
4.2 Left Invariant Vector Fields
148(6)
4.3 The Lie Algebra of a Lie Group
154(7)
4.3.1 Basic Properties; The Adjoint Representation
154(3)
4.3.2 From Lie Groups to Lie Algebras
157(1)
4.3.3 From Lie Algebras to Lie Groups
158(3)
4.4 A Digression on Topological Groups
161(6)
4.5 Commutative Lie Groups
167(4)
4.5.1 A Structure Theorem
167(3)
4.5.2 Towards Elliptic Curves
170(1)
4.6 Homogeneous Spaces
171(5)
4.7 Comments
176(2)
4.8 Exercises
178(7)
Chapter 5 Differential Forms
185(50)
5.1 Introduction
185(2)
5.1.1 Why Differential Forms?
185(1)
5.1.2 Abstract
186(1)
5.2 Multilinear Algebra
187(7)
5.2.1 Tensor Algebra
187(2)
5.2.2 Exterior Algebra
189(4)
5.2.3 Application: The Grassmannian of 2-Planes in 4 Dimensions
193(1)
5.3 The Case of Open Subsets of Euclidean Space
194(5)
5.3.1 Forms of Degree 1
194(2)
5.3.2 Forms of Arbitrary Degree
196(3)
5.4 Exterior Derivative
199(5)
5.5 Interior Product, Lie Derivative
204(5)
5.6 Poincare Lemma
209(4)
5.6.1 Star-Shaped Open Subsets
209(3)
5.6.2 Forms Depending on a Parameter
212(1)
5.7 Differential Forms on a Manifold
213(5)
5.8 Maxwell's Equations
218(4)
5.8.1 Minkowski Space
218(1)
5.8.2 The Electromagnetic Field as a Differential Form
219(1)
5.8.3 Electromagnetic Field and the Lorentz Group
220(2)
5.9 Comments
222(3)
5.10 Exercises
225(10)
Chapter 6 Integration and Applications
235(38)
6.1 Introduction
235(2)
6.2 Orientation: From Vector Spaces to Manifolds
237(7)
6.2.1 Oriented Atlas
237(2)
6.2.2 Volume Forms
239(3)
6.2.3 Orientation Covering
242(2)
6.3 Integration of Manifolds: A First Application
244(4)
6.3.1 Integral of a Differential Form of Maximum Degree
244(2)
6.3.2 The Hairy Ball Theorem
246(2)
6.4 Stokes's Theorem
248(9)
6.4.1 Integration on Compact Subsets
248(1)
6.4.2 Regular Domains and Their Boundary
249(4)
6.4.3 Stokes's Theorem in All of Its Forms
253(4)
6.5 Canonical Volume Form of a Submanifold of Euclidean Space
257(5)
6.6 Brouwer's Theorem
262(3)
6.7 Comments
265(1)
6.8 Exercises
266(7)
Chapter 7 Cohomology and Degree Theory
273(50)
7.1 Introduction
273(2)
7.2 De Rham Spaces
275(2)
7.3 Cohomology in Maximum Degree
277(4)
7.4 Degree of a Map
281(11)
7.4.1 The Case of a Circle
281(2)
7.4.2 Definition and Basic Properties in the General Case
283(3)
7.4.3 Invariance of the Degree under Homotopy: Applications
286(3)
7.4.4 Index of a Vector Field
289(3)
7.5 Fundamental Theorem of Algebra: Revisited
292(3)
7.5.1 Two Proofs of the Fundamental Theorem of Algebra Using Degree Theory
292(1)
7.5.2 Comparison of the Different Proofs of the Fundamental Theorem of Algebra
293(2)
7.6 Linking
295(4)
7.7 Invariance under Homotopy
299(4)
7.8 The Mayer-Vietoris Sequence
303(7)
7.8.1 Exact Sequences
303(1)
7.8.2 The Mayer-Vietoris Sequence
304(3)
7.8.3 Application: A Few Cohomology Calculations
307(2)
7.8.4 The Noncompact Case
309(1)
7.9 Integral Methods
310(3)
7.10 Comments
313(2)
7.11 Exercises
315(8)
Chapter 8 Euler-Poincare and Gauss-Bonnet
323(26)
8.1 Introduction
323(3)
8.1.1 From Euclid to Carl-Friedrich Gauss and Pierre-Ossian Bonnet
323(2)
8.1.2 Sketch of a Proof of the Gauss-Bonnet Theorem
325(1)
8.1.3 Abstract
326(1)
8.2 Euler-Poincare Characteristic
326(5)
8.2.1 Definition; Additivity
326(2)
8.2.2 Tilings
328(3)
8.3 Invitation to Riemannian Geometry
331(5)
8.4 Poincare-Hopf Theorem
336(3)
8.4.1 Index of a Vector Field: Revisited
336(1)
8.4.2 A Residue Theorem
336(3)
8.5 From Poincare-Hopf to Gauss-Bonnet
339(5)
8.5.1 Proof Using the Classification Theorem for Surfaces
339(1)
8.5.2 Proof Using Tilings: Sketch
340(1)
8.5.3 Putting the Preceding Arguments Together
341(3)
8.6 Comments
344(2)
8.7 Exercises
346(3)
Appendix: The Fundamental Theorem of Differential Topology 349(2)
Solutions to the Exercises 351(32)
Bibliography 383(10)
Index 393
Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs.
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