Conjugate points on Lie groups with left-invariant metrics

TL;DR

This paper establishes conditions for the existence of conjugate points along geodesics in Lie groups with left-invariant metrics, providing bounds and criteria applicable to compact, noncompact, and homogeneous cases, with implications for geometry and physics.

Contribution

It introduces new criteria and bounds for conjugate points on Lie groups with left-invariant metrics, extending previous curvature-based methods and applying to important cases like SU(n) and SO(n).

Findings

All geodesics in compact semisimple Lie groups have conjugate points.

Provides bounds on conjugate times for these geodesics.

Establishes criteria for conjugate points in noncompact and homogeneous cases.

Abstract

We prove sufficient conditions for the existence of conjugate points along geodesics of a left-invariant metric on a Lie group, using a reformulation of the index form in terms of the adjoint action. In the compact semisimple case, with an arbitrary left-invariant metric, we show that all geodesics must have a conjugate point, and we give upper and lower bounds on conjugate times. In particular this applies to the left-invariant metrics on $SU(n)$ and $SO(n)$ which are of importance in fluid dynamics and rigid body motion, and yields estimates for the diameter and injectivity radius. We also establish criteria in the noncompact case: we show that every closed nonhomogeneous geodesic has a conjugate point, and determine explicit conditions for them in the three-dimensional unimodular case. For homogeneous geodesics, we relate conjugate points to Lagrangian stability, and Eulerian stability of the corresponding steady velocity. Finally, we obtain as by-products criteria for conjugate points in general homogeneous spaces, by lifting the problem to the total Lie group of the quotient and using a result of O'Neill. Through several examples, we show that our theorems apply when well-known criteria relying on positive Ricci curvature or other curvature bounds fail, and in some cases even when Ricci curvature is negative in all directions.