Resumen de Euler characteristics for Gaussian fields on manifolds

Jonathan E. Taylor, Robert J. Adler

• 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 \Ee[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 \Ee[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 \Pp[supp∈Mf(p)≥u]. Our proofs rely on results from random fields on \Rrn as well as differential and Riemannian geometry.