Let $M$ be a finite dimensional manifold and $k:M\times M\rightarrow\mathbb{R}$ a smooth reproducing kernel. Is the set of all such $k$ with the property that the reproducing kernel Hilbert space generated by $k$ is infinite dimensional an open subset of the collection of all smooth reproducing kernels?
More generally, are there any references that investigate how the set of reproducing kernels on a manifold is organized according to RKHS dimension?