We work on a closed smooth Riemannian manifold $(M,g)$ and let $K:M\times M\to \mathbb R\cup\{+\infty\}$ be a kernel, which we assume to be integrable and lower semicontinuous. We say it is positive definite (modifying definitions a bit, so that singular kernels are allowed) if $$ \int_M\int_M K(x,y) f(x) f(y)d\mathrm{vol}_g(x)\ d\mathrm{vol}_g(y) \ge 0 $$ holds for all $f\in C^\infty(M)$.
I am interested to know cases when it is known if (or if not) the kernel $K(d_g(x,y))$ is positive definite, in which $d_g$ is the geodesic distance.
For example, I don't even know this for the "Riesz kernel formula" analogue $1/(d_g(x,y))^s$ on the torus of dimension $d$, for $0<s<d$ (I suspect it may be false).
Often for kernels on manifolds people don't take the formulas for kernels that work on $\mathbb R^d$, rather they find analogues that are positive definite by construction, so I had trouble finding any known cases in the literature. (However I did not know what to search for.)
