Geodesically Equivalent Metrics on Homogenous Spaces

TL;DR

This paper proves that on homogeneous spaces, geodesically equivalent invariant metrics are affinely equivalent, and provides an algorithm to find such metrics, demonstrating their existence and non-existence in specific cases.

Contribution

It establishes that geodesically equivalent invariant metrics are affinely equivalent and introduces an algorithm to identify all such metrics on Lie groups.

Findings

Geodesically equivalent invariant metrics are affinely equivalent.

No two left invariant metrics on S^3 are geodesically equivalent.

Examples of Lie groups with geodesically equivalent, non-proportional metrics are provided.

Abstract

Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that existence of non-proportional, geodesically equivalent, $G$-invariant metrics on homogenous space $G/H$ implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metric, of any signature, on sphere $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, non-proportional, left-invariant metrics.