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Separation of variables for Riemannian spaces of constant curvature. (English) Zbl 0658.53041
Pitman Monographs and Surveys in Pure and Applied Mathematics, 28. Harlow, Essex: Longman Scientific and Technical; New York: John Wiley & Sons, Inc.; VIII, 172 p.; \$ 56.95 (1986).

The main purpose of this book is to give a complete answer, for a class of Riemannian manifolds, to the problem: how many “inequivalent” coordinate systems does a manifold admit which give a complete integral of the Hamilton-Jacobi equation, having an additive-separable form? The author also gives the notion of product separation, which occurs for a Riemannian manifold when one seeks solutions of the Helmholtz equation with a potential $V=0$, having a product-separable form. This class of manifolds consists of the real positive definite Riemannian manifolds of constant curvature. The problem of the classification of all “inequivalent” separable coordinate systems on a given Riemannian manifold is solved completely for the real n-sphere ${S}^{n}$, the upper sheet of the double-sheeted hyperboloid ${H}^{n}$ and the real Euclidean n-space ${E}^{n}$ for the Hamilton-Jacobi and Helmholtz equations.

The other principal problems arising in the mathematical theory of separation of variables are outlined, namely, how is it possible to characterize in a coordinate-free geometric way the occurence of additive or product separation of variables, given a Riemannian manifold M, and, what are the “inequivalent” types of additive or product separation of variables that can occur on a Riemannian manifold of dimension n? Constructions and properties of separable coordinate systems are given. The interplay between group theory and the constraints of separation of variables theory enables an elegant solution to be obtained.

The graphical calculus thereby developed is extended, first, to give a complete classification of all inequivalent separable coordinate systems for Laplace’s equation and the null Hamilton-Jacobi equation on conformally Euclidean n-space ${E}^{n}$, and, second, to give the classification of all “R-separable” coordinate systems for the heat equation of ${E}^{n}·$

Other aspects of the theory of separation of variables are mentioned, viz. the generalizations of classification of “inequivalent” coodinate systems to complex Riemannian manifolds, the relationship between the functions of mathematical physics and Lie group theory, the intrinsic characterization of separation of variables, the development of a mathematical theory for separation of variable technique applied to the non-scalar-valued equations of mathematical physics, etc. Finally, the author emphasizes that much of this work is a consequence of original research done in collaboration with W. Miller jun.

##### MSC:
 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global) 53-02 Research monographs (differential geometry) 58-02 Research monographs (global analysis) 35A25 Other special methods (PDE) 58J60 Relations of PDE with special manifold structures 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)