Geodesically equivalent metrics on homogenous spaces
Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two -invariant metrics of arbitrary signature on homogenous space 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, -invariant metrics on homogenous space implies that their holonomy algebra cannot be full. We give an algorithm for...