Geodesic
Geodesically equivalent metrics on homogenous spaces
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 945-954.

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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.
DOI : 10.21136/CMJ.2018.0557-17
Classification : 22E15, 53C22, 53C30
Keywords: invariant metric; geodesically equivalent metric; affinely equivalent metric 
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Bokan, Neda; Šukilović, Tijana; Vukmirović, Srdjan. Geodesically equivalent metrics on homogenous spaces. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 4, pp. 945-954. doi : 10.21136/CMJ.2018.0557-17. http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0557-17/

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