I have recently heard about the existence of the so called metallic Riemannian manifolds. As far as I understand, those are manifold with a polynomial structure, compatible with the Riemannian metric, induced by a $(1,1)$ tensor $J$ satisfying a metallic equation $$ J^2 = pJ + qI $$ where $p$ and $q$ are positive integers and $I$ is the identity operator.
Why are they studied? Is there any geometric application? I mean, does the existence of a metallic structure imply some interesting geometric or topological property of the manifold?
I've never heard of them.
I don't think there are constraints for a metric to be metallic. Suppose $M,g$ is a Riemannian manifold and chose $p,q$ positive and integer. Let $\lambda_{\pm}=\frac{p\pm\sqrt{p^2+4q}}{2}$. Consider $J=\lambda_{\pm} I$. Then $J^2=pJ+qI$ and I believe $(M,g,J)$ is metallic.