Аннотация:
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.
This is joint work with N. Bokan and S. Vukmirović .
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