Riemannian cubics in quadratic matrix Lie groups

Abstract

Quadratic matrix Lie groups are subgroups of the general linear group that satisfy a quadratic matrix identity. The main purpose of this paper is to consider Riemannian cubics in quadratic matrix Lie groups with left-invariant metrics. Results for Riemannian cubics in quadratic matrix Lie groups extend those in SO(n) since the group SO(n) with bi-invariant metric is a very special case. By examining Riemannian cubics in SO(2, 1) and SO(3, 1), we find that the so-called null Lie quadratics in so(p,q) (p > 0, q > 0), and even more generally for any quadratic matrix Lie group, can be given in closed forms in terms of Lie quadratics in so(p) and so(q). Further, we present some quantitative analyses of non-null Lie quadratics in so(p,q).