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
Author(s): Andreas Arvanitogeorgos
Series: Student Mathematical Library, Vol. 22
Publisher: American Mathematical Society
Year: 2003
Language: English
Pages: C+xvi+141+B
Cover
S Title
An Introduction to Lie Groups and the Geometry of Homogeneous Spaces
© 2003 by the American Mathematical Society
ISBN 0-8218-2778-2
QA387.A78 2003 512'.55-dc22
LCCN 2003058352
Contents
Preface
Introduction
Chapter 1 Lie Groups
1. An example of a Lie group
2. Smooth manifolds: A review
3. Lie groups
4. The tangent space of a Lie group - Lie algebras
6. The Campbell-Baker-Hausdorff formula
7. Lie's theorems
Chapter 2 Maximal Tori and the Classification Theorem
1. Representation theory: elementary concepts
2. The adjoint representation
3. The Killing form
4. Maximal tori
5. The classification of compact and connected Lie groups
6. Complex semisimple Lie algebras
Chapter 3 The Geometry of a Compact Lie Group
1. Riemannian manifolds: A review
2. Left-invariant and bi-invariant metrics
3. Geometrical aspects of a compact Lie group
Chapter 4 Homogeneous Spaces
1. Coset manifolds
2. Reductive homogeneous spaces
3. The isotropy representation
Chapter 5 The Geometry of a Reductive Homogeneous Space
1. G-invariant metrics
2. The Riemannian connection
3. Curvature
Chapter 6 Symmetric Spaces
1. Introduction
2. The structure of a symmetric space
3. The geometry of a symmetric space
4. Duality
Chapter 7 Generalized Flag Manifolds
1. Introduction
2. Generalized flag manifolds as adjoint orbits
3. Lie theoretic description of a generalized flag manifold
4. Painted Dynkin diagrams
5. T-roots and the isotropy representation
6. G-invariant Riemannian metrics
7. G-invariant complex structures and Kahler metrics
8. G-invariant Kahler-Einstein metrics
9. Generalized flag manifolds as complex manifolds
Chapter 8 Advanced topics
1. Einstein metrics on homogeneous spaces
Isotropy irreducible spaces
Normal homogeneous spaces
Einstein metrics on generalized flag manifolds
2. Homogeneous spaces in symplectic geometry
A classical Hamiltonian system.
A Hamiltonian system on generalized flag manifolds.
3. Homogeneous geodesics in homogeneous spaces
Low-dimensional examples
Bibliography
Index
Back Cover
