Euler characteristics for Gaussian fields on manifolds

Abstract

We are interested in the geometric properties of real-valued Gaussian random fields defined on manifolds. Our manifolds, M, are of class C3 and the random fields f are smooth. Our interest in these fields focuses on their excursion sets, f-1 [u, + ∞), and their geometric properties. Specifically, we derive the expected Euler characteristic E[χ(f-1 [u, + ∞))] of an excursion set of a smooth Gaussian random field. Part of the motivation for this comes from the fact that E[χ(f-1[u, + ∞))] 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 P[supp∈M f(p) ≥ u]. Our proofs rely on results from random fields on ℝn as well as differential and Riemannian geometry.