The ladder of Finsler-type objects and their variational problems on spacetimes

The space of anisotropic $r$-contravariant $s$-covariant $\alpha$-homogeneous tensors on a manifold admits a functorial structure where vertical derivatives $\dot{\mathrm{\partial }}$ and contractions ${\mathrm{ı}}_{\mathbb{C}}$ by the Liouville vector field $\mathbb{C}$ are operators which maintain $s+\alpha$ constant. In (semi-)Finsler geometry, this structure is transmitted faithfully to connection-type elements yielding the following ladder: geodesic sprays / nonlinear connections / anisotropic connections / linear (Finslerian) connections. However, it is more loosely transmitted to metric-type ones: Finslerian Lagrangians / Legendre transformations / anisotropic metrics. We will study this structure in depth and apply it to discuss the recent variational proposals (Einstein-Hilbert, Einstein-Palatini, Einstein-Cartan) for generalizing Einstein equations to the Finsler setting.