Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 469-495, 2017
On the projective Finsler metrizability and the integrability of Rapcsák equation
Tamás Milkovszki, Zoltán Muzsnay
Received January 10, 2016. First published May 12, 2017.
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.