**Student Mathematical Library**

Volume: 22;
2003;
148 pp;
Softcover

MSC: Primary 53; 22; 17;

Print ISBN: 978-0-8218-2778-9

Product Code: STML/22

List Price: $34.00

Individual Price: $27.20

**Electronic ISBN: 978-1-4704-2136-6
Product Code: STML/22.E**

List Price: $34.00

Individual Price: $27.20

#### Supplemental Materials

# An Introduction to Lie Groups and the Geometry of Homogeneous Spaces

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*Andreas Arvanitoyeorgos*

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.

#### Table of Contents

# Table of Contents

## An Introduction to Lie Groups and the Geometry of Homogeneous Spaces

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface ix10 free
- Introduction xi12 free
- Chapter 1. Lie Groups 118 free
- Chapter 2. Maximal Tori and the Classification Theorem 2340
- Chapter 3. The Geometry of a Compact Lie Group 5168
- Chapter 4. Homogeneous Spaces 6582
- Chapter 5. The Geometry of a Reductive Homogeneous Space 7794
- Chapter 6. Symmetric Spaces 87104
- Chapter 7. Generalized Flag Manifolds 95112
- 1. Introduction 95112
- 2. Generalized flag manifolds as adjoint orbits 96113
- 3. Lie theoretic description of a generalized flag manifold 98115
- 4. Painted Dynkin diagrams 98115
- 5. T-roots and the isotropy representation 100117
- 6. G-invariant Riemannian metrics 103120
- 7. G-invariant complex structures and Kahler metrics 105122
- 8. G-invariant Kahler-Einstein metrics 108125
- 9. Generalized flag manifolds as complex manifolds 111128

- Chapter 8. Advanced topics 113130
- Bibliography 129146
- Index 139156
- Back Cover Back Cover1161

#### Readership

Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, topology, harmonic analysis, and mathematical physics.