The method of separation of variables for Laplace-Beltrami equation in semi-Riemannian geometry

Abstract

Let M be a semi-Riemannian manifold with boundary ∂M (possibly ∂M = O). If the metric on M is indefinite then the Laplace-Beltrami equation Δf= 0 on M is an ultra-hyperbolic type equation. Hence even if M is compact and ∂M = O, unlike the Riemannian case, Δf = 0 may have nonconstant solutions. In Partial Differential Equations, a common method of solving this equation is the method of separation of variables. But if one examines this method, it can be seen that the domain of solution is given as a product manifold and then the spectral theory for Laplacian is applied. The purpose of this note is to generalize this method to double-warped semi-Riemannian manifolds, which is the maximal extension of this method of finding solutions. As in Partial Differential Equations, when we choose the factor manifolds Riemannian and impose the Dirichlet or Neumann boundary conditions on M, the Laplace-Beltrami equation decomposes to elliptic, self-adjoint operators on factor manifolds. Then by making use of the spectral theory of such operators, we will obtain some solutions of the Laplace-Beltrami equation. We will also define the generalized wave equation and show that there exists a unique solution for that equation for the usual wave boundary conditions. Intuitively, the generalized wave equation may be considered that “drum” is changing it’s shape in time.