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 ) .