Is a valid correlation function on $R^2$ (a plane) also a valid correlation function on a cylinder Suppose to have a valid correlation function on $R^2$ that depends on the distance. For example if  the distance between two point is $h$, the correlation function can be $\exp(-\phi h)$, where $\phi$ is a parameter.
Now if instead of  $R^2$ we are on a cylinder: $R \times D$ where $D$ is a circle. My question is:
The correlation function based on the distance is again a valid correlation function on the cylinder?
Or maybe the question can be expressed in a different way: how can I prove that a correlation function on a generic space is a valid correlation function?
 A: I answer my own question:
If we move from a covariance function of a linear domain, $R^d$ to a cylinder, we have to prove (i.e. check if it is positive definite) if it is a valid covariance function unless we think to the cylinder as a object in $R^3$ and then we use the distance in $R^3$.
Unfortunately i am not sure i could post some examples on the cylinder because we are still working on it, and the results will be published. 
But we can ask a different, but similar, question: Is a covariance function on $\mathbb{R}$ be valid also in $\mathbb{S}$. There are two paper that answer to this question: "When is a truncated covariance function on the line a covariance function on the circle? Andrew T.A. Wood" and "Simple tests for the validity of correlation function models on the circle" of Gneiting. 
As an example (taken to the Gneiting paper) the covariance function: $\frac{1}{2}\left(\exp( -t^2)+(1+3|t|)(  1-|t|)_+^3   \right)$
where $t$ is the distance and $(  1-|t|)_+^3 = (  1-|t|)^3$ if $(  1-|t|)^3>0$ and 0 otherwise, is not positive definite in a circle.
