This book provides an introduction to several aspects of the geometry of Lie groups and homogenous spaces. Although detailed proofs for some theorems in the book are given, there are several theorems that are stated without proof. Therefore, the book presents some advanced materials at a quick pace for graduate students and research mathematicians. We believe that the reader can benefit from this book if he/she has some prior knowledge of Manifolds. An excellent prerequisite reading for this book is the book titled Topology of Surfaces, Knots, and Manifolds written by Stephan C. Carlson.
The book consists of eight chapters. The first chapter is on Lie Groups. In this chapter, the author presents a very short account of smooth manifolds and then defines the concept of a Lie group. He then provides a large set of examples for Lie groups. Among other concepts defined in this chapter are: The tangent space of a Lie group, Lie algebras, and infinitesimal groups. The author states The Campbell-Baket-Hausdorff formula without proving it. Likewise, he presents two additional key theorems that describe the relation between Lie groups and their Lie algebras without proof.
The second chapter is on Maximal Tori and the Classification Theorem. The author proves the first and second versions of Schur’s Lemma and shows that any finite-dimensional representation of a compact group is a direct sum of irreducible representations. After defining the concepts of adjoint representation, the killing form, and maximal tori, he classifies compact connected Lie groups. This chapter ends with a discussion of complex semisimple Lie groups.
The third chapter is on the Geometry of a Compact Lie Group. The chapter starts with a brief review of Riemannian manifolds. Two examples are given. He proves the first and the second Bianchi identities and shows that there is a one-to-one correspondence between bi-invariant metrics on a Lie group and Ad-invariant products on its Lie algebra. The chapter ends with a discussion of geometric aspects of a compact Lie group.
The fourth chapter is on Homogeneous Spaces. The author starts this chapter with the definition of Coset manifolds and presents, without proof, a proposition regarding the quotient group G/K, where G is a Lie group and K is a closed subgroup of G. This proposition states that there is a unique way to make G/K a manifold so that the projection from G to G/K is a submersion. Next, he defines the notion of a Reductive homogeneous space along with some examples. The chapter ends with a discussion of isotropy representation.
The fifth chapter is on the Geometry of Reductive Homogeneous Spaces. The concepts of a G-invariant metric, Riemannian connection, and curvature are discussed. Several examples are given.
Chapter six, which is on Symmetric Spaces, starts with the definition of a locally symmetric Riemannian manifold. Then a theorem by Cartan which provides a necessary and sufficient condition (in terms of the curvature of the curvature tensor) for a Riemannian manifold to be locally symmetric is presented. Next, the author defines symmetric Riemannian manifolds and presents some of the properties of these manifolds.
The seventh chapter is on Generalized Flag Manifolds. These are homogeneous spaces of the form G/C(S), where G is a compact Lie group, and C(S) is the centralizer of a torus S in G. He presents Generalized flag manifolds as adjunct orbits and as complex manifolds. The chapter contains discussions of Painted Dynkin Diagrams, T-roots and the Isotropy Representations, and G-invariant Kahler-Einstein Metrics.
Chapter eight is on more advanced topics such as Einstein Metrics on Homogeneous Spaces, Einstein Metrics on Generalized Flag Manifolds, and Homogeneous Geodesics in Homogeneous Spaces.
Overall, this book is an excellent short survey of materials that span over fields such as differential geometry, algebraic topology, harmonic analysis, and mathematical physics. This book is an excellent resource for graduate students and research mathematicians.
Morteza Seddighin is an associate professor of mathematics at Indiana University East. His research interests are functional analysis and operator theory.