Let $M$ be an arbitrary finite-dimensional smooth manifold. For simplicity, let's assume that $M$ has no boundary. Does there always exist a gaussian random field with constant variance on $M$? If not, does there exist a theorem which states sufficient (or necessary and sufficient) conditions on $M$ under which a gaussian random field will always exist?

My definition of a gaussian random field is a random function $f:M\to \mathbb{R}^{n}$ such that for arbitrary points $t_1, \dots, t_k$ in $M$, the images $f(t_1),\dots, f(t_k)$ form a multivariate guassian random vector. Adler, Taylor, and Worsley in Applications of Random Fields and Geoemtry assume that $f(t)$ has constant variance for all $t$.