On the projective Finsler metrizability and the integrability of Rapcsák equation

Milkovszki, Tamás and Muzsnay, Zoltán (2017) On the projective Finsler metrizability and the integrability of Rapcsák equation. CZECHOSLOVAK MATHEMATICAL JOURNAL, 67 (2). pp. 469-495. ISSN 0011-4642

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Abstract

A. Rapcsak 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 Rapcsak system and the extended Rapcsak system, by using the Spencer version of the Cartan-Kahler theorem. We also consider the extended Rapcsak 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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