# Does every smooth manifold carry a gaussian random field?

Let $M$ be an arbitrary smooth manifold of dimension $n$. For simplicity, let's assume that $M$ is boundary-less. Can we construct a gaussian random field on $M$?

If the result is not true for arbitrary $M$, is there a theorem that provides minimal assumptions on $M$ for which we always can find a gaussian random field?

• The existence is certainly true for any manifold: just take a constant field. – zhoraster Oct 21 '15 at 6:15