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-definitereal n×n matrices SPD(n), including necessary and sufficent conditions for a submanifold N ⊂ SPD(n) to be totally geodesic for the affine-invariant Riemannian metric.A non-linear projection x → π(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 ∈ SPD(n) to the unique closest element π(x) in thetotally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space SPD(n) follow, as well as variants of thepolar decomposition of non-singular matrices known as Mostow’s decompositions.Applications to decompositions of covariant matrices are mentioned.