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Critical point theory and submanifold geometry. (English) Zbl 0658.49001
Lecture Notes in Mathematics, 1353. Berlin etc.: Springer-Verlag. x, 272 p. DM 42.50 (1988).
The book provides a modern introduction into the geometry of submanifolds of Euclidean as well as of Hilbert spaces and to the critical point theory on Hilbert manifolds; both topics are closely related through the Morse index theorem. The geometrical content is unusually comprehensive. Besides the clear and concise general theory and a wealth of short examples and exercises, various rather concrete topics are thoroughly discussed. We can mention the Weingarten surfaces in three-dimensional space (including the umbilic surfaces, Bäcklund transforms and Bianchi compositions for the constant Gauss curvature), flat tori, applications of Coxeter groups and Dynkin diagrams, the Mountain Pass Theorem, the Ljusternik-Schnirelman theorem, harmonic maps, and the Yamabe problem. In this respect, the book belongs to the most interesting expositions of differential geometry which has ever been written.
The authors assume the familiarity with the elements of differential manifolds and Riemannian geometry but the introductory parts recall the connections, covariant derivatives, fundamental forms, Gauss-Codazzi equations and focal points of submanifolds by using Cartan’s moving frame method. The more advanced parts are directed towards the classification and topology of the so called isoparametric submanifolds (the normal bundle of which is flat and the principal curvatures along parallel normal vector fields are constant) arising (for instance) as the orbits of certain transformation groups. The used tools are however thoroughly discussed and of independent interest. They include the theory of isometric Fredholm actions on Hilbert manifolds, Fredholm submanifolds, Coxeter groups and especially the Morse theory.
The latter theme is quite separately treated in a third of the book and includes (besides the standard deformation theory, attaching of handles, the minimax principle and the Morse inequalities) also a refined theory of linking cycles useful for effective calculation of homologies of isoparametric submanifolds. A brief review of the variational calculus with several non-trivial applications of the previous general principles conclude this beautiful book.
Reviewer: J.Chrastina

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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