Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds

TL;DR

This paper investigates the Jacobi osculating rank of geodesics on naturally reductive 3-manifolds, revealing that most have rank two, with specific exceptions, and characterizes isotropic geodesics and conjugate loci.

Contribution

It provides a detailed analysis of the Jacobi osculating rank in naturally reductive 3-manifolds and classifies isotropic geodesics and conjugate loci, extending geometric understanding.

Findings

Jacobi osculating rank is two for all geodesics except Hopf fibers

All isotropic geodesics are explicitly determined

The isotropic tangent conjugate locus is characterized

Abstract

We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of its horizontal distribution is also a constant. Then, we prove that the Jacobi osculating rank of every geodesic is two except for the Hopf fibers, where it is zero. Moreover, we determine all isotropic geodesics and the isotropic tangent conjugate locus.