Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices

Abstract: This paper is a self-contained exposition of the geometry of symmetric positive-definite real $n\times n$ matrices $\operatorname{SPD}(n)$, including necessary and sufficent conditions for a submanifold $\mathcal{N} \subset\operatorname{SPD}(n)$ to be totally geodesic for the affine-invariant Riemannian metric. A non-linear projection $x\mapsto \pi(x)$ on a totally geodesic submanifold is defined. This projection has the minimizing property with respect to the Riemannian metric: it maps an arbitrary point $x \in\operatorname{SPD}(n)$ to the unique closest element $\pi(x)$ in the totally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space $\operatorname{SPD}(n)$ follow, as well as variants of the polar decomposition of non-singular matrices known as Mostow's decompositions. Applications to decompositions of covariant matrices are mentioned.