The study of homogeneous spaces provides excellent insights into both
differential geometry and Lie groups. In geometry, for instance, general
theorems and properties will also hold for homogeneous spaces, and will
usually be easier to understand and to prove in this setting. For Lie
groups, a significant amount of analysis either begins with or reduces
to analysis on homogeneous spaces, frequently on symmetric spaces. For
many years and for many mathematicians, Sigurdur Helgason's classic
Differential Geometry, Lie Groups, and Symmetric Spaces has been--and
continues to be--the standard source for this material.
<P>Helgason begins with a concise, self-contained introduction to
differential geometry. He then introduces Lie groups and Lie algebras,
including important results on their structure. This sets the stage for
the introduction and study of symmetric spaces, which form the central
part of the book. The text concludes with the classification of
symmetric spaces by means of the Killing-Cartan classification of simple
Lie algebras over $\mathbf{C}$ and Cartan's classification of simple
Lie algebras over $\mathbf{R}$. <P>The excellent exposition is
supplemented by extensive collections of useful exercises at the end of
each chapter. All the problems have either solutions or substantial
hints, found at the back of the book. <P>For this latest edition,
Helgason has made corrections and added helpful notes and useful
references. The sequels to the present book are published in the AMS's
Mathematical Surveys and Monographs Series: Groups and Geometric
Analysis, Volume 83, and Geometric Analysis on Symmetric Spaces, Volume
39. <P>Sigurdur Helgason was awarded the Steele Prize for
Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and
Geometric Analysis.
