Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 469–495

On the projective Finsler metrizability and the integrability of Rapcsák equation



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.


Euler-Lagrange equation metrizability projective metrizability geodesics spray formal integrability 

MSC 2010

49N45 58E30 53C60 53C22 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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