Geodesics, curvature, and conjugate points on Lie groups

Abstract

In a Lie group equipped with a left-invariant metric, we study the minimizing properties of geodesics through the presence of conjugate points. We give criteria for the existence of conjugate points along steady and nonsteady geodesics, using different strategies in each case. We consider both general Lie groups and quadratic Lie groups, where the metric in the Lie algebra $g(u,v)=\langle u,\Lambda v\rangle$ is defined from a bi-invariant bilinear form and a symmetric positive definite operator $\Lambda$. By way of illustration, we apply our criteria to $SO(n)$ equipped with a generalized version of the rigid body metric, and to Lie groups arising from Cheeger's deformation technique, which include Zeitlin's $SU(3)$ model of hydrodynamics on the $2$-sphere. Along the way we obtain formulas for the Ricci curvatures in these examples, showing that conjugate points occur even in the presence of some negative curvature.