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

This paper is a self-contained exposition of the geometry of symmetric positive-definite real n× n matrices SPD(n), including necessary and sufficent conditions for a submanifold N (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 (n) to the unique closest element π(x) in the totally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space 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.