Question: is there a "generalized" Sasaki metric for Finsler manifolds?
More directly, let $(M,F)$ be a Finsler manifold with the Cartan connection. Is there a Finsler manifold $(TM,\hat{F})$ such that if $(M,F)$ is actually Riemannian, then $\hat{F}$ reduces to the Sasaki metric?
If there is more than one, which might not surprise me, which is considered "natural" and why?
So far, I've found some sparse literature (e.g. Finsler Geometry and Natural Foliations on the Tangent Bundle by Bejancu and Farran), but would like others' thoughts.
