A ninth printing of Helgason's 1978 textbook and reference published by Academic Press, itself a revision of and sequel to his 1962 Differential Geometry and Symmetric Spaces , based in turn on lectures he gave a the University of Chicago in 1958 and later at Columbia and MIT. He begins by explaining differential geometry, emphasizing Riemannian geometry, then applies that to the basic theory of Lie groups and Lie algebras. A two-volume sequence, Groups and Geometric Analysis and Geometric Analysis on Symmetric Spaces have been published in the Society's series Mathematical Surveys and Monographs. Annotation c. Book News, Inc., Portland, OR (booknews.com)