Introduction to Geometry of Manifolds with Symmetry

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1 Introduction to Geometry of Manifolds with Symmetry

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 270

3 Introduction to Geometry of Manifolds with Symmetry by v. v. Trofimov Depanment of Geometry and Topology, Moscow State University, Moscow, Russia SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress Cataloging-in-Publication Data Trof1mov, V. V., [Vveden1e v geometr1fu mnogoobrazi1 s s1mmetrifam1. Engl1shl Introduct1on ta geometry of man1folds w1th symmetry 1 by V.V. Trofimov. p. cm. -- <Mathemat1cs and its appl1cat1ons ; v. 270) Includes b1b11ograph1cal references and index. ISBN ISBN (ebook) DOI / Geometry. 2. Man1folds<Mathemat1cs) 3. Symmetry. I. 11tle. II. Ser1es: Mathemat1cs and its appl1cations <Kluwer Academic Publ1shersl ; v QA447.T '. 07--dc ISBN This book was typeset using AMSTeX This is an updated translation of the original work Introduction to Geometry of Manifolds with Symmetries, Moscow, Moscow University Press Translated by G. G. Okuneva Printed on acid-free paper Ali Rights Reserved 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 CONTENTS Notation Preface Chapter I. Elements of differential geometry 1. The notion of a topological space 2. Continuous mappings of topological spaces 3. Count ability axioms 4. Quasi-compact topological spaces 5. Separation axioms 6. Sequentially compact topological spaces 7. Construction of topological spaces 8. Smooth manifolds 9. Geometry of smooth manifolds 10. Elements of tensor algebra 11. Smooth mappings of smooth manifolds 12. Exterior differential forms on manifolds 13. Integration of exterior differential forms 14. De Rham cohomology 15. Elements of Riemannian geometry 16. Elements of affine geometry 17. Curvature tensor 18. Geodesics and the shortest Chapter II. Lie Groups and Lie Algebras 1. Lie groups 2. Lie algebras 3. Trajectories of left-invariant vector fields 4. The exponential mapping 5. Displacement of functions along trajectories 6. Actions of Lie groups 7. Linear representations of Lie groups 8. Automorphisms of Lie groups 9. Maurer-Cartan formula 10. Basic global theorems about Lie groups 11. Problems of non-simple connectedness. Coverings 12. Lie subgroups 13. Nilpotent representations of Lie algebras 14. Solvable Lie algebras and their linear representations vii ix v

6 vi CONTENTS 15. Representations of nilpotent Lie algebras Semisimple Lie algebras Cartan subalgebras Killing metric Cartan's criterion Structure of semisimple Lie algebras Simple Lie algebras Analyticity 154 Chapter III. Symmetric spaces Notion of a symmetric space Compact Lie groups as Riemannian symmetric spaces Involute automorphisms of Lie groups and related Riemannian symmetric spaces Connections in principal bundles Basic theorems Lie groups as symmetric spaces Totally geodesic submanifolds Totally geodesic submanifolds and involute automorphisms Riemannian symmetric spaces 219 Chapter IV. Smooth vector bundles and characteristic classes Vector bundles Connections and metrics in bundles Covariant derivation and curvature Characteristic classes of vector bundles Basic characteristic classes Connectedness structures in principal bundles of frames Transgression The Euler class Geometric sense of the Euler class in dimension two Geometric sense of the Euler class in higher dimensions 271 Chapter V. Applications Commutation equations for differential operators Poisson brackets of hydrodynamic type and left-symmetric algebras Differential equations of motion of rigid body about a fixed point Compatible Poisson brackets Invariants of coadjoint representation 314 Bibliography 321 Index 323

7 NOTATION lr C Q xea AcB AnB AUB A\B AxB CA AE9B I:A--+B Iml I(A) I-l(y) ker 1 log, or Ig a/\b TzM COO(M) lr n En V* sn lrpn DX/dt Vxy DX V kx 8s = 8/8x s the set of reals the set of complex numbers the set of rationals x is an element of A A is a subset of B the intersection of A and B the union of A and B the difference of A and B (the part of A that is not in B the Cartesian product of A and B the complement of A the direct sum of linear spaces A and B 1 is a mapping of A to B the image of a mapping 1 the image of a set A under a mapping 1 the preimage of a point y under a mapping 1 (the set of all x with I(x) = y) the kernel of a linear operator 1 (the preimage of zero) the composition of 1 and 9 the exterior product of differential forms a and b the tangent space of a manifold M at a point x the space of smooth functions on a manifold M the n-dimensional arithmetic space the n-dimensional Euclidean space the dual space of a linear space V the n-dimensional sphere; the subset of lr n +1 determined by the equation (xl ) (xn+1? = 1 the n-dimensional real projective space the covariant derivative of X with respect to t the covariant derivative of Y in the direction of X the covariant differential the covariant partial derivative the usual partial derivative The "method of root letters and indices" is used: the root letter a in the symbol ai' indicates that the symbol is related to the object a; the index i runs through the values 1,2,3,... ; the prime at the index i is related to the place where the index stands. Thus, the places for indices (but not the indices themselves) may be either not primed, or primed (possibly, several times). A the closure of a set A vii

8 viii A' p(x, y) dett trt Hk(M) SL(n,lR) O(n) U(n) r~k R;,pq S}k exp vol(d) X NOTATION the set of limit points of A the distance between x and y the determinant of a matrix A the trace of a matrix T the k-dimensional cohomology group of a manifold M the set of real matrices whose determinants are equal to one the set of orthogonal matrices the set of unitary matrices the Christoffel symbols the curvature tensor the torsion tensor the exponential mapping the volume of a domain D the derivative of x with respect to t

9 PREFACE One ofthe most important features of the development of physical and mathematical sciences in the beginning of the 20th century was the demolition of prevailing views of the three-dimensional Euclidean space as the only possible mathematical description of real physical space. Apriorization of geometrical notions and identification of physical space with its mathematical modellr 3 were characteristic for these views. The discovery of non-euclidean geometries led mathematicians to the understanding that Euclidean geometry is nothing more than one of many logically admissible geometrical systems. Relativity theory amended our understanding of the problem of space by amalgamating space and time into an integral four-dimensional manifold. One of the most important problems, lying at the crossroad of natural sciences and philosophy is the problem of the structure of the world as a whole. There are a lot of possibilities for the topology offourdimensional space-time, and at first sight a lot of possibilities arise in cosmology. In principle, not only can the global topology of the universe be complicated, but also smaller scale topological structures can be very nontrivial. One can imagine two "usual" spaces connected with a "throat", making the topology of the union complicated. A leading role in various topological constructions belongs to the notion of symmetry. Since antiquity, symmetry is recognized as an attribute of beauty. Ancient books presumed that the universe is symmetric simply because symmetry is beautiful. The idea of symmetry often served as a guiding thread in attempts to solve problems about the structure of the universe. Let us mention, for example, regular polyhedra. Regular geometrical figures have long since been attracting attention of mathematicians, astronomers and artists. In the eyes of some of them, the figures were surrounded with a mystical aura and seemed to be involved with mysteries of the world. Thus, the famous German astronomer of the end of the 16th century Iohann Kepler presumed that the structure of the Solar system can be explained from a geometrical scheme. Imagine a sphere with the center at the sun and the radius equal to the one of the orbit of Saturn; inscribe a cube in this sphere. Then the orbit of Jupiter lies in the sphere inscribed in this cube. If we inscribe a regular tetrahedron into this sphere, then the sphere inscribed in it will contain the orbit of Mars. One should proceed the inscription, placing a dodecahedron between Mars and Earth, an icosahedron between Earth and Venus, and an octahedron between Venus and Mercury. Though, Kepler indicated that the numeric relations do not match precisely. Later, his construction was ruined by the discovery of Uranus, Neptune and Pluto. The idea of classification of geometries with the help of symmetries was realized in 1872 by F. Klein. According to the Erlangen program of Klein, a geometry is ix

10 x PREFACE characterized by the set of transformations of geometrical objects assumed admissible in this geometry, as well as by properties of the objects that are invariant under the admissible transformations. A geometry is determined by the group of its transformations. The most well-known geometries, the Euclidean geometry, the affine geometry, and the projective geometry provide good examples for this approach. In Euclidean geometry, figures are compared via movements of the plane or the space. One can apply translations, rotations, or reflections with respect to a line or a plane to produce the family of all figures equal to a given one. In affine geometry, the set of transformations is wider: one may use any transformation that take straight lines onto straight lines, parallel to parallel. Projective geometry studies the properties of figures that are preserved by a still wider class of transformations, the so-called projective transformations. The most general transformation group arises in topology: here one can bend figures, or continuously deform them; only cutting into pieces and gluing from pieces is forbidden. In physics, symmetry reveals itself in the conservation laws. The energy preservation law reflects the symmetry with respect to translations in time. The metric is homogeneous in time, that is, no distinguished moments of time exist. The momentum preservation law (three components) is connected with the symmetry of the theory with respect to translations in space. The metric of space is homogeneous, i.e., no distinguished points exist in space. The angular momentum preservation law (three components) is connected with the symmetry of the theory with respect to rotations of the space. The metric of space is isotropic, that is, no distinguished directions exist. Preservation of the center of gravity is connected with the symmetry of the theory with respect to the "Lorentzian rotation" in space-time, that is, with the space-time isotopy. The electric charge preservation law comes from a more complicated symmetry. It is a consequence of the symmetry with respect to gauge transformations. Another type of symmetries of physical objects is connected with the so-called inner symmetries of elementary particles. A mathematical description of these requires a more complicated language of vector bundles over manifolds. The inner symmetry reveals, in particular, in the fact that the particles unite into families (multiplets). The particles in the same multiplet have almost equal masses. Each multiplet realizes an irreducible representation of the group of inner symmetries. This book is an introduction to the geometry of manifolds equipped with additional structures, connected with the notion of symmetry. The book is based on materials of special courses read by the author for students of the Department of Mechanics and Mathematics in the Moscow State University. In Chapter 1, a preliminary one, we expose several topics in differential geometry, necessary for the sequel. The first part of this chapter (Sections 1-7) contains some necessary facts from general topology. The second part (Sections 8-14) is an introduction to the theory of smooth manifolds. Here we discuss such notions as tangent vectors, tensor fields, trajectories, differential forms and their integrals over manifolds. The rest sections constitute the third part, devoted to manifolds with additional structures: affine connectedness and Riemannian metric. We give here an introduction to tensor calculus on manifolds. Chapter 2 is devoted to basic notions of the theory of Lie groups and Lie algebras. Its first part (Sections 1-12) contains an introduction to Lie groups. We discuss

11 PREFACE xi here such notions as Lie groups and their Lie algebras, action of a Lie group on a manifold, and a representation of a Lie group. In the second part (Sections 13-21) we introduce basic notions of the theory of Lie algebras. The exposition comes to a study of the structure of semisimple Lie algebras over algebraically closed fields (this topic is applied in recent investigations in the theory of completely integrable Hamiltonian systems). Chapter 3 is devoted to basic notions of the theory of symmetric spaces. We present both the theory of Riemannian and affinely connected symmetric spaces. One of the main topics here is the realization of affinely connected symmetric spaces as totally geodesic submanifolds of Lie groups. In Chapter 4, we construct the most important characteristic classes of vector bundles. The construction is carried out in terms of differential forms; we do not present the axiomatic approach here. We study especially closely the geometry of the Euler class. Chapter 5 presents some applications of the previous general geometrical notions and constructions. In particular, we give an introduction to modern methods for integration of nonlinear differential equations, discovered and studied by S. P. Novikov and his school. We consider here some questions of the theory of hydrodynamic type Poisson brackets and show some connections with interesting algebraic structures. We also give a geometrical view of the so called coordinated Poisson brackets and an introduction to the theory of integration of differential equations on Lie algebras. The book is intended for a wide range of readers and requires only minimal starting knowledge of mathematics. It suffices, for example, to know linear algebra as exposed in [10] and differential equations as in [17]. The book is accessible for second-year students in mathematics and mechanics. I wish to express my gratitude to my teachers and colleagues, who helped me to appreciate the beauty of geometry. I am especially indebted to Professor A. T. Fomenko, who did much for my becoming a mathematician. I wish to thank the Kluwer Academic Publishers and Professor M. Hazewinkel and Doctor D. J. Larner for their support and interest in this work. I am also grateful to G. G. Okuneva for much effort she put in the translation of this book.

Differential Geometry, Lie Groups, and Symmetric Spaces

Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason Graduate Studies in Mathematics Volume 34 nsffvjl American Mathematical Society l Providence, Rhode Island PREFACE PREFACE TO THE