An algebraic characterisation for Finsler metrics of constant flag curvature

In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators in Finsler geometry, Bucataru and Muzsnay (Symmetry Integr Geom Methods Appl 7:114, 2011), Grifone and Muzsnay (Variational principles for second-order differential equations. World Scientific, Singapore, 2000). This algebraic characterisation, for Finsler metrics of constant flag curvature, allows to provide yet another proof for the Finslerian version of Beltrami's theorem, Bucataru and Cretu (J Geom Anal 30:617-631, 2020; Publ Math Debr,, 2019).

Automated summary

This paper proves the algebraic relationship between the constant flag curvature of a Finsler metric and the curvature of the induced nonlinear connection, which serves as an obstacle to the formal integrability of operators in Finsler geometry. Furthermore, this algebraic characterization provides another proof for the Finslerian version of Beltrami's theorem.