Subriemannian Metrics and the Metrizability of Parabolic Geometries

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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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