This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.
The first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations, connections, Riemannian manifolds, Lie groups, and Hamiltonian mechanics. Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.
This second edition greatly expands upon the first by including more examples, additional exercises, and new topics, such as the moment map and fiber bundles. Detailed solutions to every exercise are also provided.
Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and a basic knowledge of analytical mechanics
Author(s): Gerardo F. Torres del Castillo
Edition: 2
Publisher: Birkhäuser
Year: 2020
Language: English
Pages: 444
Tags: Manifolds, Lie Derivatives, Differential Forms, Connections, Lie Groups, Hamiltonian Mechanics
Preface to the Second Edition
Preface to the First Edition
Contents
1 Manifolds
1.1 Differentiable Manifolds
1.2 The Tangent Space
1.3 Vector Fields
1.4 1-Forms and Tensor Fields
2 Lie Derivatives
2.1 One-Parameter Groups of Transformations and Flows
2.2 Lie Derivative of Functions and Vector Fields
2.3 Lie Derivative of 1-Forms and Tensor Fields
3 Differential Forms
3.1 The Algebra of Forms
3.2 The Exterior Derivative
4 Integral Manifolds
4.1 The Rectification Lemma
4.2 Distributions and the Frobenius Theorem
4.3 Symmetries and Integrating Factors
5 Connections
5.1 Covariant Differentiation
5.2 Torsion and Curvature
5.3 The Cartan Structural Equations
5.4 Tensor-Valued Forms and Covariant Exterior Derivative
6 Riemannian Manifolds
6.1 The Metric Tensor
6.2 The Riemannian Connection
6.3 Curvature of a Riemannian Manifold
6.4 Volume Element, Divergence, and Duality of Differential Forms
6.5 Elementary Treatment of the Geometry of Surfaces
7 Lie Groups
7.1 Basic Concepts
7.2 The Lie Algebra of the Group
7.2.1 The Structure Constants
7.2.2 Lie Group Homomorphisms
7.2.3 The SU(2)-SO(3) Homomorphism
7.2.4 Lie Subgroups
7.3 Invariant Differential Forms
7.3.1 The Maurer–Cartan Equations
7.3.2 Finding the Group from the Structure Constants
7.3.3 Invariant Forms on Subgroups of GL(n, mathbbR)
7.4 One-Parameter Subgroups and the Exponential Map
7.4.1 A Torsion-Free Connection
7.5 The Lie Algebra of the Right-Invariant Vector Fields
7.6 Lie Groups of Transformations
7.6.1 The Adjoint Representation
7.6.2 The Coadjoint Representation
8 Hamiltonian Classical Mechanics
8.1 The Cotangent Bundle
8.2 Hamiltonian Vector Fields and the Poisson Bracket
8.3 The Phase Space and the Hamilton Equations
8.4 Geodesics, the Fermat Principle, and Geometrical Optics
8.5 Dynamical Symmetry Groups
8.6 The Moment Map
8.7 The Rigid Body and the Euler Equations
8.8 Time-Dependent Formalism
Appendix A Lie Algebras
Appendix B Invariant Metrics
Appendix C Fiber Bundles
Appendix Solutions
Exercises of Chap.1摥映數爠eflinkchap:111
Exercises of Chap.2摥映數爠eflinkchap:222
Exercises of Chap.3摥映數爠eflinkchap:333
Exercises of Chap.4摥映數爠eflinkchap:444
Exercises of Chap.5摥映數爠eflinkchap:555
Exercises of Chap.6摥映數爠eflinkchap:666
Exercises of Chap.7摥映數爠eflinkchap:777
Exercises of Chap.8摥映數爠eflinkchap:888
Exercises of Appendix A
Exercises of Appendix B
Exercises of Appendix C
Appendix References
Index
