An Introduction to Lie Groups and the Geometry of Homogeneous Spaces (Student Mathematical Library, V. 22)

It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics: algebra, differential geometry, algebraic geometry, analysis, and differential equations. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry of Lie groups and homogeneous spaces. In this way, he avoids the extra detail needed for a thorough discussion of representation theory. Lie groups and homogeneous spaces are especially useful to study in geometry, as they provide excellent examples where quantities (such as curvature) are easier to compute. A good understanding of them provides lasting intuition, especially in differential geometry. The author provides several examples and computations. Topics discussed include the classification of compact and connected Lie groups, Lie algebras, geometrical aspects of compact Lie groups and reductive homogeneous spaces, and important classes of homogeneous spaces, such as symmetric spaces and flag manifolds. Applications to more advanced topics are also included, such as homogeneous Einstein metrics, Hamiltonian systems, and homogeneous geodesics in homogeneous spaces. The book is suitable for advanced undergraduates, graduate students, and research mathematicians interested in differential geometry and neighboring fields, such as topology, harmonic analysis, and mathematical physics. Readership: Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, topology, harmonic analysis, and mathematical physics Cover 1 S Title 2 An Introduction to Lie Groups and the Geometry of Homogeneous Spaces 4 © 2003 by the American Mathematical Society 5 ISBN 0-8218-2778-2 5 QA387.A78 2003 512'.55-dc22 5 LCCN 2003058352 5 Contents 6 Preface 10 Introduction 12 Chapter 1 Lie Groups 18 1. An example of a Lie group 18 2. Smooth manifolds: A review 19 3. Lie groups 25 4. The tangent space of a Lie group - Lie algebras 29 6. The Campbell-Baker-Hausdorff formula 37 7. Lie's theorems 38 Chapter 2 Maximal Tori and the Classification Theorem 40 1. Representation theory: elementary concepts 41 2. The adjoint representation 45 3. The Killing form 49 4. Maximal tori 53 5. The classification of compact and connected Lie groups 56 6. Complex semisimple Lie algebras 58 Chapter 3 The Geometry of a Compact Lie Group 68 1. Riemannian manifolds: A review 68 2. Left-invariant and bi-invariant metrics 76 3. Geometrical aspects of a compact Lie group 78 Chapter 4 Homogeneous Spaces 82 1. Coset manifolds 82 2. Reductive homogeneous spaces 88 3. The isotropy representation 89 Chapter 5 The Geometry of a Reductive Homogeneous Space 94 1. G-invariant metrics 94 2. The Riemannian connection 96 3. Curvature 97 Chapter 6 Symmetric Spaces 104 1. Introduction 104 2. The structure of a symmetric space 105 3. The geometry of a symmetric space 108 4. Duality 109 Chapter 7 Generalized Flag Manifolds 112 1. Introduction 112 2. Generalized flag manifolds as adjoint orbits 113 3. Lie theoretic description of a generalized flag manifold 115 4. Painted Dynkin diagrams 115 5. T-roots and the isotropy representation 117 6. G-invariant Riemannian metrics 120 7. G-invariant complex structures and Kahler metrics 122 8. G-invariant Kahler-Einstein metrics 125 9. Generalized flag manifolds as complex manifolds 128 Chapter 8 Advanced topics 130 1. Einstein metrics on homogeneous spaces 130 Isotropy irreducible spaces 131 Normal homogeneous spaces 132 Einstein metrics on generalized flag manifolds 133 2. Homogeneous spaces in symplectic geometry 135 A classical Hamiltonian system. 136 A Hamiltonian system on generalized flag manifolds. 137 3. Homogeneous geodesics in homogeneous spaces 140 Low-dimensional examples 143 Bibliography 146 Index 156 Back Cover 160 "It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics: algebra, differential geometry, algebraic geometry, analysis, and differential equations. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry of Lie groups and homogeneous spaces. In this way, he avoids the extra detail needed for a thorough discussion of representation theory. Lie groups and homogeneous spaces are especially useful to study in geometry, as they provide excellent examples where quantities (such as curvature) are easier to compute. A good understanding of them provides lasting intuition, especially in differential geometry. The author provides several examples and computations. Topics discussed include the classification of compact and connected Lie groups, Lie algebras, geometrical aspects of compact Lie groups and reductive homogeneous spaces, and important classes of homogeneous spaces, such as symmetric spaces and flag manifolds. Applications to more advanced topics are also included, such as homogeneous Einstein metrics, Hamiltonian systems, and homogeneous geodesics in homogeneous spaces."--Provided by publisher

It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry of Lie groups and homogeneous spaces. In this way, he avoids the extra detail needed for a thorough discussion of other topics. Lie groups and homogeneous spaces are especially useful to study in geometry, as they provide excellent examples where quantities (such as curvature) are easier to compute. A good understanding of them provides lasting intuition, especially in differential geometry. The book is suitable for advanced undergraduates, graduate students, and research mathematicians interested in differential geometry and neighboring fields, such as topology, harmonic analysis, and mathematical physics.

A Lie group is a set that has both a manifold and a group structure, which are compatible.