Variational aspects of homogeneous geodesics on generalized flag manifolds and applications

TL;DR

This paper investigates conjugate points along homogeneous geodesics in generalized flag manifolds using second variation analysis and explores how homogeneous Ricci flow can generate conjugate points in complex projective spaces.

Contribution

It introduces a variational approach to study conjugate points on homogeneous geodesics and demonstrates the evolution of such points under Ricci flow in specific manifolds.

Findings

Conjugate points can be characterized via second variation analysis.

Ricci flow can induce conjugate points in complex projective spaces.

The approach applies to generalized flag manifolds and related geometries.

Abstract

We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space $\mathbb{C} P^{2n+1} = Sp(n+1)/({U}(1)\times{Sp}(n))$.