Geodesically equivalent metrics on homogenous spaces. (English) Zbl 07144866

The present paper is an interesting study on geodesically equivalent metrics. Two metrics on a (pseudo) Riemannian manifold are geodesically equivalent if the sets of their unparametrized geodesics coinside. The main result shows that if two \(G\)-invariant metrics on a homogeneous space \(G/H\) are geodesically equivalent, then they are affinely equivalent, i.e., they have the same Levi-Civita connection. A method is provided in order to find all left-invariant metrics geodesically equivalent to a given left-invariant metric, on a Lie group. As a consequence, it is shown that no two left-invariant metrics of any signature on the sphere \(\mathbb{S}^3\) are geodesically equivalent. Also, examples of Lie groups are provided, that admit geodesically equivalent, nonproportional, left-invariant metrics. Another result of the paper proves that, the existence of nonproportional, geodesically equivalent, \(G\)-invariant metrics on \(G/H\), implies that their holonomy algebra cannot be full.


53C22 Geodesics in global differential geometry
22E15 General properties and structure of real Lie groups
53C30 Differential geometry of homogeneous manifolds
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