Differential geometry on matrix groups

Abstract

In this dissertation, we study the differential geometry of the matrix groups. In literatures, many authors proposed to compute matrix means using the geodesic distance defined by an invariant Riemannian metrics on the matrix groups, and this approach has become predominant. Most of papers in this field deal with either the space of positive definite matrices or some special matrix group, and their results have a quite similar form. The geometric means of the unitary group U(n) was proposed by Mello in 1990. Later, Moakher gave the geodesic means on special orthogonal group SO(3) and symmetric positive definite matrix space SPD(n). In Chapter 1, the explicit geodesic and gradient forms of special matrix groups are given, which would provide theoretical basis of computing the geometric mean. We have also presented the results in a more unified form than those that have appeared in the current literature.

Then, we study the curvatures of the matrix groups. Because the curvature provides important information about the geometric structure of a Riemannian manifold. For example, it is related to the rate at which two geodesics emitting from the same point move away from each other: the lower the curvature is, the faster they move apart. Many important geometric and topological properties are implied by suitable curvature conditions. In Chapter 2, we give a simple formula for sectional curvatures on the general linear group, which is also valid for many other matrix groups. This formula appears to be new in literature and is extended to more general reductive Lie groups. Additionally, we also discuss the relation between commuting matrices and zero sectional curvature for GL(n, R).