Czechoslovak Mathematical Journal, first online, pp. 1-10

Geodesically equivalent metrics on homogenous spaces

Neda Bokan, Tijana Šukilović, Srdjan Vukmirović

Received December 6, 2017.   Published online November 9, 2018.

Abstract:  Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$-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 the existence of nonproportional, 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 metrics of any signature on sphere $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.
Keywords:  invariant metric; geodesically equivalent metric; affinely equivalent metric
Classification MSC:  53C22, 22E15, 53C30
DOI:  10.21136/CMJ.2018.0557-17

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Affiliations:   Neda Bokan, Tijana Šukilović, Srdjan Vukmirović, University of Belgrade, Faculty of Mathematics, Studentski trg 16, Belgrade 11000, Serbia, e-mail:,,

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