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Topics in Differential Geometry This Page Intentionally Left Blank Topic S in Differentia L Geometr Y http://dx.doi.org/10.1090/gsm/093 Topics in Differential Geometry This page intentionally left blank Topic s in Differentia l Geometr y Peter W. Michor Graduate Studies in Mathematics Volume 93 ^j^flll American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 53-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-93 Library of Congress Cataloging-in-Publication Data Michor, Peter W., 1949- Topics in differential geometry / Peter W. Michor. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 93) Includes bibliographical references and index. ISBN 978-0-8218-2003-2 (alk. paper) 1. Geometry, Differential. I. Title. QA641.M49 2008 516.3'6—dc22 2008010629 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08 To the ladies of my life, Elli, Franziska, and Johanna This page intentionally left blank Contents Preface CHAPTER I. Manifolds and Vector Fields 1. Differentiate Manifolds 2. Submersions and Immersions 3. Vector Fields and Flows CHAPTER II. Lie Groups and Group Actions 4. Lie Groups I 5. Lie Groups II. Lie Subgroups and Homogeneous Spaces 6. Transformation Groups and G-Manifolds 7. Polynomial and Smooth Invariant Theory CHAPTER III. Differential Forms and de Rham Cohomology 8. Vector Bundles 9. Differential Forms 10. Integration on Manifolds 11. De Rham Cohomology 12. Cohomology with Compact Supports and Poincare Duality 13. De Rham Cohomology of Compact Manifolds 14. Lie Groups III. Analysis on Lie Groups 15. Extensions of Lie Algebras and Lie Groups Vlll Contents CHAPTER IV. Bundles and Connections 16. Derivations on the Algebra of Differential Forms 17. Fiber Bundles and Connections 18. Principal Fiber Bundles and G-Bundles 19. Principal and Induced Connections 20. Characteristic Classes 21. Jets CHAPTER V. Riemann Manifolds 22. Pseudo-Riemann Metrics and Covariant Derivatives 23. Geometry of Geodesies 24. Parallel Transport and Curvature 25. Computing with Adapted Frames and Examples 26. Riemann Immersions and Submersions 27. Jacobi Fields CHAPTER VI. Isometric Group Actions or Riemann G-Manifolds 28. Isometries, Homogeneous Manifolds, and Symmetric Spaces 29. Riemann G-Manifolds 30. Polar Actions CHAPTER VII. Symplectic and Poisson Geometry 31. Symplectic Geometry and Classical Mechanics 32. Completely Integrable Hamiltonian Systems 33. Poisson Manifolds 34. Hamiltonian Group Actions and Momentum Mappings List of Symbols Bibliography Index Preface This book is an introduction to the fundamentals of differential geometry (manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann mani• folds, isometric actions, symplectic geometry) which stresses naturality and functoriality from the beginning and is as coordinate-free as possible. The material presented in the beginning is standard — but some parts are not so easily found in text books: Among these are initial submanifolds (2.13) and the extension of the Probenius theorem for distributions of nonconstant rank (the Stefan-Sussman theory) in (3.21) - (3.28). A quick proof of the Campbell-Baker-Hausdorff formula for Lie groups is in (4.29). Lie group actions are studied in detail: Palais' results that an infinitesimal action of a finite-dimensional Lie algebra on a manifold integrates to a local action of a Lie group and that proper actions admit slices are presented with full proofs in sections (5) and (6). The basics of invariant theory are given in section (7): The Hilbert-Nagata theorem is proved, and Schwarz's theorem on smooth invariant functions is discussed, but not proved. In the section on vector bundles, the Lie derivative is treated for natural vector bundles, i.e., functors which associate vector bundles to manifolds and vector bundle homomorphisms to local diffeomorphisms. A formula for the Lie derivative is given in the form of a commutator, but it involves the tangent bundle of the vector bundle. So also a careful treatment of tangent bundles of vector bundles is given. Then follows a standard presentation of differential forms and de Rham cohomoloy including the theorems of de Rham and Poincare duality. This is used to compute the cohomology of compact Lie groups, and a section on extensions of Lie algebras and Lie groups follows. IX X Preface The chapter on bundles and connections starts with a thorough treatment of the Frolicher-Nijenhuis bracket via the study of all graded derivations of the algebra of differential forms. This bracket is a natural extension of the Lie bracket from vector fields to tangent bundle valued differential forms; it is one of the basic structures of differential geometry. We begin our treatment of connections in the general setting of fiber bundles (without structure group). A connection on a fiber bundle is just a projection onto the vertical bundle. Curvature and the Bianchi identity are expressed with the help of the Frolicher-Nijenhuis bracket. The parallel transport for such a general connection is not defined along the whole of the curve in the base in general — if this is the case, the connection is called complete. We show that every fiber bundle admits complete connections. For complete connections we treat holonomy groups and the holonomy Lie algebra, a subalgebra of the Lie algebra of all vector fields on the standard fiber. Then we present principal bundles and associated bundles in detail together with the most important examples. Finally we investigate principal connections by requiring equivariance under the structure group. It is remarkable how fast the usual structure equations can be derived from the basic properties of the Frolicher-Nijenhuis bracket. Induced connections are investigated thoroughly — we describe tools to recognize induced connections among general ones. If the holonomy Lie algebra of a connection on a fiber bundle with compact standard fiber turns out to be finite-dimensional, we are able to show that in fact the fiber bundle is associated to a principal bundle and the connection is an induced one. I think that the treatment of connections presented here offers some didactical advantages: The geometric content of a connection is treated first, and the additional requirement of equi variance under a structure group is seen to be additional and can be dealt with later — so the student is not required to grasp all the structures at the same time. Besides that it gives new results and new insights. This treatment is taken from [146]. The chapter on Riemann geometry contains a careful treatment of connec• tions to geodesic structures to sprays to connectors and back to connections considering also the roles of the second and third tangent bundles in this. Most standard results are proved. Isometric immersions and Riemann sub• mersions are treated in analogy to each other. A unusual feature is the Jacobi flow on the second tangent bundle. The chapter on isometric ac• tions starts off with homogeneous Riemann manifolds and the beginnings of symmetric space theory; then Riemann G-manifolds and polar actions are treated. The final chapter on symplectic and Poisson geometry puts some emphasis on group actions, momentum mappings and reductions. Preface XI There are some glaring omissions: The Laplace-Beltrami operator is treated only summarily, there is no spectral theory, and the structure theory of Lie algebras is not treated and used. Thus the finer theory of symmetric spaces is outside of the scope of this book. The exposition is not always linear. Sometimes concepts treated in detail in later sections are used or pointed out earlier on when they appear in a natural way. Text cross-references to sections, subsections, theorems, numbered equations, items in a list, etc., appear in parantheses, for example, section (1), subsection (1.1), theorem (3.16), equation (3.16.3) which will be called (3) within (3.16) and its proof, property (3.22.1). This book grew out of lectures which I have given during the last three decades on advanced differential geometry, Lie groups and their actions, Riemann geometry, and symplectic geometry. I have benefited a lot from the advise of colleagues and remarks by readers and students. In particular I want to thank Konstanze Rietsch whose write-up of my lecture course on isometric group actions was
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