The index in semi-Riemannian geometry and classification of partial differential equations

Semi-Riemannian geometry studies manifolds with a nonsingular metric tensor. Two cases have been studied in detail: Riemannian geometry, when the tensor is positive-definite, and Lorentz geometry, when at each point of the manifold it is negative-definite in a subspace of dimension 1 of the tangent space.

Writing the metric in diagonal form, its index is the number of negative diagonal terms. The Laplace operator Δ on a manifold, the divergence of the gradient of a function, has as many negative terms as this index. The Laplace equation Δv = 0 is elliptic in Riemannian geometry and hyperbolic in a Lorentz geometry. The ultrahyperbolic case when the index is neither 0 nor 1 has been much less studied. But this relationship between semi-Riemannian geometry and partial differential equations is not often mentioned in the literature.