Geodesically Equivalent Metrics on Homogenous Spaces
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...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.1805.08240 |