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 $$ .