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Differentiable Manifolds Gerardo F. Torres del Castillo Differentiable Manifolds ATheoreticalPhysicsApproach Gerardo F. Torres del Castillo Instituto de Ciencias Universidad Autónoma de Puebla Ciudad Universitaria 72570 Puebla, Puebla, Mexico [email protected] ISBN 978-0-8176-8270-5 e-ISBN 978-0-8176-8271-2 DOI 10.1007/978-0-8176-8271-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011939950 Mathematics Subject Classification (2010): 22E70, 34C14, 53B20, 58A15, 70H05 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Preface The aim of this book is to present in an elementary manner the basic notions related with differentiable manifolds and some of their applications, especially in physics. The book is aimed at advanced undergraduate and graduate students in physics and mathematics, assuming a working knowledge of calculus in several variables, linear algebra, and differential equations. For the last chapter, which deals with Hamilto- nian mechanics, it is useful to have some previous knowledge of analytical mechan- ics. Most of the applications of the formalism considered here are related to dif- ferential equations, differential geometry, and Hamiltonian mechanics, which may serve as an introduction to specialized treatises on these subjects. One of the aims of this book is to emphasize the connections among the areas of mathematics and physics where the formalism of differentiable manifolds is applied. The themes treated in the book are somewhat standard, but the examples developed here go beyond the elementary ones, trying to show how the formalism works in actual calculations. Some results not previously presented in book form are also in- cluded, most of them related to the Hamiltonian formalism of classical mechanics. Whenever possible, coordinate-free definitions or calculations are presented; how- ever, when it is convenient or necessary, computations using bases or coordinates are given, not underestimating their importance. Throughout the work there is a collection of exercises, of various degrees of difficulty, which form an essential part of the book. It is advisable that the reader attempt to solve them and to fill in the details of the computations presented in the book. The basic formalism is presented in Chaps. 1 and 3 (differentiable manifolds, differentiable mappings, tangent vectors, vector fields, and differential forms), after which the reader, if interested in applications to differential geometry and general relativity, can continue with Chaps. 5 and 6 (even though in the definitions of a Killing vector field and of the divergence of a vector field given in Chap. 6,the definition of the Lie derivative, presented in Chap. 2,isrequired).Chapter7 deals with Lie groups and makes use of concepts and results presented in Chap. 2 (one- parameter groups and Lie derivatives). Chapters 2 and 4 are related with differential equations and can be read in an independent form, after Chaps. 1 and 3.Finally, v vi Preface for Chap. 8,whichdealswithHamiltonianmechanics,thematerialofChaps.1, 2, and 3,isnecessaryand,forsomesections,Chaps.6 and 7 are also required. Some of the subjects not treated here are the integration of differential forms, cohomology theory, fiber bundles, complex manifolds, manifolds with boundary, and infinite-dimensional manifolds. This book has been gradually developed starting from a first version in Spanish (with the title Notas sobre variedades diferenciables)writtenaround1981,atthe Centro de Investigación y de Estudios Avanzados del IPN, in Mexico, D.F. The previous versions of the book have been used by the author and some colleagues in courses addressed to advanced undergraduate and graduate students in physics and mathematics. IwouldliketothankGilbertoSilvaOrtigoza,MercedMontesinos,andthere- viewers for helpful comments, and Bogar Díaz Jiménez for his valuable help with the figures. I also thank Jessica Belanger, Tom Grasso, and Katherine Ghezzi at Birkhäuser for their valuable support. Puebla, Puebla, Mexico Gerardo F. Torres del Castillo Contents 1Manifolds................................. 1 1.1 Differentiable Manifolds . 1 1.2 The Tangent Space .......................... 9 1.3 Vector Fields . 17 1.4 1-Forms and Tensor Fields . 21 2LieDerivatives.............................. 29 2.1 One-Parameter Groups of Transformations and Flows ....... 29 2.2 Lie Derivative of Functions and Vector Fields . ......... 39 2.3 Lie Derivative of 1-Forms and Tensor Fields . 42 3DifferentialForms............................ 49 3.1 The Algebra of Forms ........................ 49 3.2 The Exterior Derivative . 55 4IntegralManifolds............................ 67 4.1 The Rectification Lemma . 67 4.2 Distributions and the Frobenius Theorem .............. 71 4.3 Symmetries and Integrating Factors . 83 5Connections................................ 93 5.1 Covariant Differentiation . 93 5.2 Torsion and Curvature . 100 5.3 The Cartan Structural Equations . 104 5.4 Tensor-Valued Forms and Covariant Exterior Derivative . 110 6RiemannianManifolds..........................115 6.1 The Metric Tensor . 115 6.2 The Riemannian Connection . ...................131 6.3 Curvature of a Riemannian Manifold ................142 6.4 Volume Element, Divergence, and Duality of Differential Forms . 149 6.5 Elementary Treatment of the Geometry of Surfaces ........154 vii viii Contents 7LieGroups................................161 7.1 Basic Concepts ............................161 7.2 The Lie Algebra of the Group . ...................166 7.3 Invariant Differential Forms . 177 7.4 One-Parameter Subgroups and the Exponential Map ........184 7.5 The Lie Algebra of the Right-Invariant Vector Fields ........190 7.6 Lie Groups of Transformations ...................192 8HamiltonianClassicalMechanics....................201 8.1 The Cotangent Bundle ........................201 8.2 Hamiltonian Vector Fields and the Poisson Bracket . 205 8.3 The Phase Space and the Hamilton Equations . .........211 8.4 Geodesics, the Fermat Principle, and Geometrical Optics .....218 8.5 Dynamical Symmetry Groups . ...................224 8.6 The Rigid Body and the Euler Equations . 236 8.7 Time-Dependent Formalism . ...................245 Appendix A Lie Algebras ..........................255 Appendix B Invariant Metrics .......................257 References ...................................269 Index ......................................271 Chapter 1 Manifolds The basic objective of the theory of differentiable manifolds is to extend the appli- n cation of the concepts and results of the calculus of the R spaces to sets that do not possess the structure of a normed vector space. The differentiability of a function n m of R to R means that around each interior point of its domain the function can be approximated by a linear transformation, but this requires the notions of linearity and distance, which are not present in an arbitrary set. The essential idea in the definition of a manifold should already be familiar from analytic geometry, where one represents the points of the Euclidean plane by a pair of real numbers (e.g., Cartesian or polar coordinates). Roughly speaking, a manifold is a set whose points can be labeled by coordinates. In this chapter and the following three, the basic formalism applicable to any finite-dimensional manifold is presented, without imposing any additional structure. In Chaps. 5 and 6 we consider manifolds with a connection and a metric tensor, respectively, which are essential in differential geometry. 1.1 Differentiable Manifolds Let M be a set. A chart (or local chart)onM is a pair (U, φ) such that U is a subset n of M and φ is a one-to-one map from U onto some open subset of R (see Fig. 1.1). AchartonM is also called a coordinate system on M.Definingachart(U, φ) on asetM amounts to labeling each point p U by means of n real numbers, since n ∈ φ(p) belongs to R ,andthereforeconsistsofn real numbers that depend on p;that is, φ(p) is of the form φ(p) x1(p), x2(p), . , xn(p) . (1.1) = This relation defines the n functions! x1,x2,...,xn,whichwillbecalledthecoordi-" nate functions or, simply coordinates, associated with the chart (U, φ).Thefactthat φ is a one-to-one mapping ensures that two different points of U differ, at least, in the value of one of the coordinates. G.F. Torres del Castillo, Differentiable Manifolds, 1 DOI 10.1007/978-0-8176-8271-2_1,©SpringerScience+BusinessMedia,LLC2012 2 1 Manifolds Fig. 1.1 AcoordinatesysteminasetM;withtheaidofφ,eachpointofU corresponds to some point of Rn.TheimageofU under φ must be an open subset of some Rn We would also like close points to have close coordinates, but that requires some notion of nearness in M,whichcanbegivenbythedefinitionofadistancebe- tween points of M or, more
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