Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Abstract

Given a homogeneous pseudo-Riemannian space \((G/H,\langle \ , \ \rangle),\) a geodesic \(\gamma :I\rightarrow G/H\) is said to be two-step homogeneous if it admits a parametrization \(t=\phi (s)\) (s affine parameter) and vectors XY in the Lie algebra \({\mathfrak{g}}\), such that \(\gamma (t)=\exp (tX)\exp (tY)\cdot o\), for all \(t\in \phi (I)\). As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics \(\langle \ ,\ \rangle\) on the unimodular Lie group \(SL(2,{{\mathbb{R}}})\) such that \(\big (SL(2,{{\mathbb{R}}}),\langle \ ,\ \rangle \big )\) is a two-step g.o. space.

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Acknowledgements

The first and third authors were supported by Grant \(\# E.037\) from the Research Committee of the University of Patras (Programme K. Karatheodori). The first author was supported by a Grant from the Empirikion Foundation in Athens. The second author was supported by funds of MIUR (within PRIN), GNSAGA and University of Salento. All authors appreciate the useful detailed comments suggested by the referees.

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Correspondence to Andreas Arvanitoyeorgos.

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Arvanitoyeorgos, A., Calvaruso, G. & Souris, N.P. Two-step homogeneous geodesics in pseudo-Riemannian manifolds. Ann Glob Anal Geom 59, 297–317 (2021). https://doi.org/10.1007/s10455-020-09751-4

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Keywords

  • Homogeneous space
  • Pseudo-Riemannian manifold
  • Homogeneous geodesic
  • Geodesic orbit space
  • Two-step homogeneous geodesic
  • Two-step geodesic orbit space
  • Generalized geodesic lemma
  • Lorentzian Lie group

Mathematics Subject Classification

  • Primary 53C22
  • Secondary 53C30
  • 53C50