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Geodesically equivalent metrics on homogenous spaces

  • Neda Bokan
  • Tijana Šukilović
  • Srdjan Vukmirović
Article
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Abstract

Two metrics on a manifold are geodesically equivalent if the sets of their un-parameterized geodesics coincide. We show that if two G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equiva-lent, 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. Us-ing that algorithm we prove that no two left invariant metrics of any signature on sphere S3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.

Keywords

invariant metric geodesically equivalent metric affinely equivalent metric 

MSC

53C22 22E15 53C30 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  • Neda Bokan
    • 1
  • Tijana Šukilović
    • 1
  • Srdjan Vukmirović
    • 1
  1. 1.University of BelgradeFaculty of MathematicsBelgradeSerbia

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