Subriemannian Metrics and the Metrizability of Parabolic Geometries

The Journal of Geometric Analysis volume 31, pages1671–1702(2021)Cite this article

Abstract

We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies. These tools lead to natural subriemannian metrics on generic distributions of interest in geometric control theory.

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Acknowledgements

The authors thank the Czech Grant Agency, Grant Nr. P201/12/G028, for financial support.

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Affiliations

  1. Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    David M. J. Calderbank

  2. Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jan Slovák

  3. Mathematical Institute, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Vladimír Souček

Authors
  1. David M. J. Calderbank
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  2. Jan Slovák
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  3. Vladimír Souček
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Corresponding author

Correspondence to Jan Slovák.

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Calderbank, D.M.J., Slovák, J. & Souček, V. Subriemannian Metrics and the Metrizability of Parabolic Geometries. J Geom Anal 31, 1671–1702 (2021). https://doi.org/10.1007/s12220-019-00320-1

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Keywords

  • Projective metrizability
  • Subriemannian metrizability
  • Weyl connections
  • Cartan geometry
  • Overdetermined linear PDE
  • Parabolic geometry
  • Bernstein–Gelfand– Gelfand resolution

Mathematics Subject Classification

  • Primary 53B15
  • 53C17
  • Secondary 14M15
  • 17B10
  • 22E46
  • 53C15
  • 53C30
  • 58A32
  • 58J70
  • 93C10