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Riemannian and Sub-Riemannian Geodesic Flows

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Abstract

We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion \(\pi :M \rightarrow B\), we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide.

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Acknowledgments

We thank Dr. Petri Kokkonen for helpful discussions and comments. The first author is partially supported by the grant Anillo CONICYT PIA ACT 1415 of the Chilean Research Council and the grant DI16-6000 of the Research and Postgraduate Studies Unit of Universidad de La Frontera. The second author is supported by Fonds National de la Recherche Luxembourg (project Open O14/7628746 GEOMREV).

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Authors and Affiliations

  1. Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile

    Mauricio Godoy Molina

  2. Mathematics Research Unit, University of Luxembourg, Luxembourg City, Luxembourg

    Erlend Grong

Authors
  1. Mauricio Godoy Molina
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  2. Erlend Grong
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Correspondence to Erlend Grong.

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Godoy Molina, M., Grong, E. Riemannian and Sub-Riemannian Geodesic Flows. J Geom Anal 27, 1260–1273 (2017). https://doi.org/10.1007/s12220-016-9717-8

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  • DOI: https://doi.org/10.1007/s12220-016-9717-8

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