Euler characteristics for Gaussian fields on manifolds

We are interested in the geometric properties of real-valued Gaussian random fields defined on manifolds. Our manifolds, $M$, are of class $C^3$ and the random fields $f$ are smooth. Our interest in these fields focuses on their excursion sets, $f^{-1}[u, +\infty)$, and their geometric properties. Specifically, we derive the expected Euler characteristic $\Ee[\chi(f^{-1}[u, +\infty))]$ of an excursion set of a smooth Gaussian random field. Part of the motivation for this comes from the fact that $\Ee[\chi(f^{-1}[u,+\infty))]$ relates global properties of $M$ to a geometry related to the covariance structure of $f$. Of further interest is the relation between the expected Euler characteristic of an excursion set above a level $u$ and $\Pp[ \sup_{p \in M} f(p) \geq u ]$. Our proofs rely on results from random fields on $\Rr^n$ as well as differential and Riemannian geometry.