Geodesically Equivalent Metrics on Homogenous Spaces
Autor: | Srdjan Vukmirovic, Tijana Sukilovic, Neda Bokan |
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Rok vydání: | 2018 |
Předmět: |
Mathematics - Differential Geometry
Pure mathematics Geodesic 010102 general mathematics Holonomy Lie group Equivalence of metrics 01 natural sciences Differential Geometry (math.DG) Ordinary differential equation FOS: Mathematics Mathematics::Differential Geometry 0101 mathematics Invariant (mathematics) Mathematics |
DOI: | 10.48550/arxiv.1805.08240 |
Popis: | 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. |
Databáze: | OpenAIRE |
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