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On the projective Finsler metrizability and the integrability of Rapcsák equation

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Abstract

A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsák system and the extended Rapcsák system, by using the Spencer version of the Cartan-Kähler theorem. We also consider the extended Rapcsák system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, Egyetem tér 1, H-4032, Debrecen, Hungary

    Tamás Milkovszki & Zoltán Muzsnay

Authors
  1. Tamás Milkovszki
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  2. Zoltán Muzsnay
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Corresponding author

Correspondence to Tamás Milkovszki.

Additional information

The research was partially supported by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreements no. 318202 and no. 317721.

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Milkovszki, T., Muzsnay, Z. On the projective Finsler metrizability and the integrability of Rapcsák equation. Czech Math J 67, 469–495 (2017). https://doi.org/10.21136/CMJ.2017.0010-16

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  • DOI: https://doi.org/10.21136/CMJ.2017.0010-16

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