On critical points of Gaussian random fields under diffeomorphic transformations.

Let and be smooth Gaussian random fields parameterized on Riemannian manifolds and , respectively, such that , where is a diffeomorphic transformation. We study the expected number and height distribution of the critical points of in connection with those of . As an important case, when is an anisotropic Gaussian random field, then we show that its expected number of critical points becomes proportional to that of an isotropic field , while the height distribution remains the same as that of .