Does every smooth manifold carry a gaussian random field? 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 $t$.
 A: You can construct   examples a dime a dozen. $\newcommand{\bR}{\mathbb{R}}$
Here  is a first  simple way. Fix $N$ smooth functions $f_1,f_2,\dotsc, f_N:M\to\bR$ and $N$ independent Gaussian random variables $X_1,\dotsc, X_N$. Then
$$f(x)=\sum_{k=1}^N X_k f_k(x) $$
is a Gaussian random field and the sample functions are a.s. smooth.
You can   allow  infinitely many functions  in the above example, i.e., $N=\infty$, but then  you need to make some assumptions on the functions and on the variances of  the variables $X_k$ to guarantee  the convergence of the resulting series. 
We know from Kolmogorov that the convergence is a $0-1$ phenomenon. The  three-series theorem  tells us what these conditions should be. (In the Gaussian case we can  say a bit more.)
Here is an example of this kind when $M$ is compact. Fix a  Riemann metric $g$ on $M$, denote by $\Delta$ the resulting Laplacian. Its eigenvalues (multiplicities included) are
$$  0=\lambda_0<\lambda_1\leq \lambda_2\leq \cdots. $$
Fix an orthonormal eigenbasis $(\psi_k)_{k\geq 0}$ of $L^2(M, dV_g)$ 
$$\Delta \psi_k=\lambda_k,\;\; \Vert\psi_k\Vert_{L^2}=1,\;\;\forall k\geq 0. $$
Next choose  independent  Gaussian random variables $(X_k)_{k\geq 0}$. We denote by $v_k$ the variance of $X_k$.   If $v_k$ goes to $0$ sufficiently fast, then the random series 
$$f(x)=\sum_{k\geq 0} X_k \psi_k(x) $$
defines a  Gaussian random field on $M$.  The regularity of the sample functions  of this random field   depends on the decay rate of  $v_k$. The faster $v_k$ decays as $k\to\infty$, the more regular is the random function. For example, if 
$$\lim_{k\to \infty}k^\alpha v_k=0,\;\;\forall \alpha>0, $$
then the random function $f(x)$ is a.s. smooth.
Ultimately, the most general construction  of a Gaussian random function on a manifold  is via Gaussian measures on the space of   distributions (i.e. generalized functions) on $M$.  I refer to   this paper  for  more details and additional references. I would start with reference [6] in this paper.
