The Einstein–Hilbert–Palatini formalism in pseudo-Finsler geometry
Miguel Ángel Javaloyes, Miguel Sánchez, Fidel F. Villaseñor
Abstract
A systematic development of the so-called Palatini formalism is carried out for pseudo-Finsler metrics $L$ of any signature. Substituting in the classical Einstein-Hilbert-Palatini functional the scalar curvature by the Finslerian Ricci scalar constructed with an independent nonlinear connection $\mathrm{N}$, the affine and metric equations for $(\mathrm{N},L)$ are obtained. In Lorentzian signature with vanishing mean Landsberg tensor Lan$_i$, both the Finslerian Hilbert metric equation and the classical Palatini conclusions are recovered by means of a combination of techniques involving the (Riemannian) maximum principle and an original argument about divisibility and fiberwise analyticity. Some of these findings are also extended to (positive definite) Riemannian metrics by using the eigenvalues of the Laplacian. When Lan$_i\neq 0$, the Palatini conclusions fail necessarily, however, a good number of properties of the solutions remain. The framework and proofs are built up in detail.