let $g=(g_{ij})$ and $h=(h_{ij})$ be two Riemannian metric on $\mathbb{R^{n}}$. consider the Riemannian metric $g\otimes h$ as a metric on $\mathbb{R}^{n^2}$.
how can we describe the geometry of the later metric in terms of two initial metrics?Is there any relation between the isometry group of the tensor metric and the isometry groups of the initial metrics?What is a relation between LC connections of these metrics?What is the result of the tensor product of two flat metrics?