The book under review is an interesting lecture notes volume based on notices of participants of a course which M. Dajczer taught during 1985/1986. The first four chapters can serve as an introduction to the differential geometry of submanifolds of arbitrary codimension in Riemannian spaces of constant sectional curvature for readers who have basic knowledge about Riemannian geometry. As special results we mention the classification of Einstein hypersurfaces of the Euclidean space, and the Jorge-Koutroufiotis theorem about Hadamard submanifolds of codimension p. Chapter 5 contains some classification results for isometric immersions with low codimension between spaces of constant sectional curvature.
The last four chapters. - 6. The theory of flat bilinear forms and isometric rigidity, 7. Conformally flat submanifolds, 8. Real Kähler submanifolds, 9. Conformal rigidity of hypersurfaces, - are devoted to the main subject of the book: the theory of J. D. Moore of flat bilinear forms and its applications. Every chapter is accompanied by a series of exercises containing interesting additional results. The bibliography contains more than a hundred items cited in the text. The book will be very useful as a textbook for a graduate course in differential geometry, and it can also be recommended to geometers working about submanifolds.