Central conjugate locus of 2-step nilpotent Lie groups

TL;DR

This paper computes the conjugate locus of central geodesics in 2-step nilpotent Lie groups with invariant metrics and compares their isometry types based on these loci, advancing understanding of their geometric structures.

Contribution

It provides explicit computation of conjugate loci for central geodesics and offers partial insights into their isometry classifications in 2-step nilpotent Lie groups.

Findings

Conjugate locus of central geodesics is explicitly computed.

Partial results on isometry type classification based on conjugate loci.

Advances understanding of geometric properties of 2-step nilpotent Lie groups.

Abstract

The goals of this article are twofold : 1) to compute the conjugate locus of a geodesic that lies in the center of a simply connected, 2-step nilpotent Lie group with a left invariant metric 2) compare the isometry types of two such nilpotent Lie groups whose conjugate loci for central geodesics are "the same" in a suitable sense. The first goal is achieved. The second is elusive, but we obtain a partial result.