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On Sub-Riemannian and Riemannian Structures on the Heisenberg Groups

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Abstract

We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate locus. Finally, we determine the minimizing geodesics from identity to any given endpoint. (Several of these points have been covered, to varying degrees, by other authors.)

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Authors and Affiliations

  1. Department of Mathematics (Pure and Applied), Rhodes University, PO Box 94, 6140, Grahamstown, South Africa

    Rory Biggs

  2. Institute of Applied Mathematics, Óbuda University, Bécsi út 96/b, H-1034, Budapest, Hungary

    Péter T. Nagy

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  1. Rory Biggs
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  2. Péter T. Nagy
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Correspondence to Rory Biggs.

Additional information

The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 317721. The first author is primarily funded by the Claude Leon Foundation.

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Biggs, R., Nagy, P.T. On Sub-Riemannian and Riemannian Structures on the Heisenberg Groups. J Dyn Control Syst 22, 563–594 (2016). https://doi.org/10.1007/s10883-016-9316-9

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