Consider the space of $n \times n$ positive definite symmetric matrices and let $\Sigma$ be one such matrix. We make this space into a Riemannian manifold $M$ by means of the metric $$ds^2=tr(\Sigma^{-1}d\Sigma\Sigma^{-1}d\Sigma)$$
Now fix a $p<n$ and consider the collection $\Phi\_p$ of all positive definite symmetric matrices which can be written
$$B^TB+D$$
for some $B$ and $D$ where $B$ is $p \times n$ and $D$ is a diagonal positive definite matrix. Here $B$ and $D$ are allowed to vary but $p$ must stay fixed. Is $\Phi\_p$ an embedded submanifold of $M$ and if so how can I put the induced metric on it? 



A submanifold of the space positive definite matrices
This Riemannian metric on the full space of all positive definite matrices turns up in the paper (and others)


* [Olga Gil-Medrano, Peter W. Michor: The Riemannian manifold of all Riemannian metrics. Quarterly J. Math. Oxford (2) 42 (1991), 183--202](http://www.mat.univie.ac.at/~michor/rie-met.pdf)


There are explicit formulas for geodesics and for curvature. There is only one incomplete geodesic, but the space if far from being geodesically convex. 


