I know that one can define metrics on the manifold of SPD matrices
$$ \mathcal{S}^n = \{ A \in \mathbb{R}^{n\times n} \ | \ \text{A positive semi-definite} \} $$
like the Log-Euclidean metric or the Riemannian Affine invariant metric (AIRM) that capture the geometry of the underlying manifold better than the standard Euclidean metric (see for example here).
I am interested in a Riemannian metric on the fixed-rank manifold
$$ \mathcal{M}_k = \{ X \in \mathbb{R}^{m\times n} \ | \ \text{rank}(X) = k \}, $$
that captures the geometry of the manifold and is also computationally accessible.
Does anybody know something? I would appreciate any help!