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First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups

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Abstract

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation H + σ = 0, where H is analogous to the mean curvature and σ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in \(\mathbb {H}^{2}\).

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Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal do Pará, Rua Augusto Corrêa, 1 - Guamá, 66075-110, Belém, PA, Brasil

    Marcos M. Diniz & José M. M. Veloso

  2. Departamento de Matemática, Universidade Federal do Amazonas, Av. General Rodrigo Octávio, 6200 - Coroado I, 69077-000, Manaus, AM, Brasil

    Maria R. B. Santos

Authors
  1. Marcos M. Diniz
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  2. Maria R. B. Santos
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  3. José M. M. Veloso
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Correspondence to Maria R. B. Santos.

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Diniz, M.M., Santos, M.R.B. & Veloso, J.M.M. First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups. J Dyn Control Syst 23, 509–533 (2017). https://doi.org/10.1007/s10883-016-9339-2

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  • DOI: https://doi.org/10.1007/s10883-016-9339-2

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