In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the wave operator $\square$. The $\square$ operator is a hyperbolic operator and $\Delta$ is an elliptic operator.
There are many results linking Riemaniann geometry through the topology of a manifold and elliptic operators. Example of this is the Atiyah-Singer theorem and Hodge-DeRham cohomology.
Are there any similar results for hyperbolic operators and Pseudo-Riemaniann geometry?