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Separation of Variables for Riemannian Spaces of Constant Curvature: E.G. Kalnins

This book arose from the desire to give a compact and, in a somewhat restricted sense, compiete treatment of the subject of separation of variables. The only book already available on the specific topic of separation of variables is that of Miller. This earlier work gives an excellent treatment of the relationships between the classical special functions of mathematical physics and Lie group theory.

The aim of the present work is to show how all the actual inequivalent separable coordinate systems can be computed for the Hamilton-Jacobi and Helmholtz equations on real positive definite Riemannian spaces of constant curvature. The results necessary for the solution of this problem are developed in the text. This allows the reader to obtain a feel for the subject without the necessity to read widely in the literature, It is in this spirit that the book has been written. Proofs that are central to the computation of all the inequivalent coordinate systems mentioned above are given in full; the more general results of the theory are often quoted, suitable references being given. We also, on occasion, appeal to the reader's intuition.

In Chapter 1 we give some introductory comments on the subject of separation of variables. Included here are the basic notions of additive and multiplicative separation of variables as well as an intuitive discussion of the basic problems of the associated theory of separation of variables. Chapter 2 sketches the historical developmentof the theory of separation of variables, providing a useful summary and extracting, from the many contributions, the most significant results. It also provides an indication of the degrees of freedom available in the specification of a separable coordinate system.

In Chapters 3, 4 and 5 we give a solution of the central problem of this work, that is, we classify the separable coordinate systems on the real n-sphere S_n , on the real Euclidean n-space E_n and on the upper sheet of the double-sheeted hyperboioid H for the Hamilton-Jacobi and Helmholtz equations. The interplay between group theory and the constraints of separation of variables theory enables an elegant solution to be obtained. The resulting graphical calculus neatly summarizes the complete solution. In Chapter 6 these methods are extended to the classification of all inequivalent separable coordinate systems for LaplaceTs equation and the null Hamilton-Jacobi equation on E_n. In Chapter 7, these ideas are further extended to the classification of all TE-separable' coordinate systems for the heat equation on E_n.

In Chapter 8 other aspects of the theory of separation of variables are mentioned:

(a) the generalization of the classification of 'inequivalent' coordinate systems to complex Eiemannian manifolds;

(b) the relationship between the special functions of mathematical physics and Lie group theory;

(c) the intrinsic characterization of separation of variables;

(d) the development of a mathematical theory for separation of variable techniques applied to the nonscalar valued equations of mathematical physics (e.g. Dirac equation, Maxwell's equations).

Much of this work is a consequence of a long-standing collaboration with my colleague Willard Miller Jr. Indeed chapters 3, 4, 5 and 6 are based on the following research reports co-authored with W, Miller Jr: Separation of variables on n dimensional manifolds

1, The n sphere S_n and Euclidean n space R_n

2, The n dimensional hyperboloid H_n

3, ConformalLy Euclidean spaces

The first of these is to be published in the Journal of Mathematical Physics, I would also like to acknowledge the influence and collaboration of Charles Boyer, Greg Reid and Pavel Winternitz. Finally, I thank my wife for her persistence in urging me to write this book.

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