On the covariant coefficients of geodesic sprays on Finsler manifolds

For a Finsler metric F, we introduce the notion of F-covariant coefficients Hᵢ of the geodesic spray of F (Def. 3.1). We study some geometric consequences concerning the objects Hᵢ. If the F-covariant coefficients Hᵢ are written in the form Hᵢ=∂̇ᵢH, for some smooth function H on T-1ptM, positively 3-homogeneous in y, then H is called spray scalar or simply S-scalar. We prove that if the S-scalar exists, then it is of the form H=1/12\,yⁱ∂ᵢF² and this expression is unique up to a function of position only. We prove also that on a Finsler maifold $(M,F)$, the S-scalar H exists if and only if $(M,F)$ is dually flat. Generally, the n³ functions Hʰᵢⱼ resulting from the F-covariant coefficients do not form a linear connection. We find out that in the case of projectively flat metrics, the n³ functions Hʰᵢⱼ are coefficients of a linear connection. We introduce two new special Finsler spaces, namely, the H-Berwald and the H-Landsberg spaces and show that every H-Berwald metric is H-Landsbergian but the converse is not necessarily true. Also, we study the F-covariant coefficients Hᵢ of projectivly flat and dually flat spherically symmetric Finsler metrics and provide a solution of the "H-unicorn" Landsberg problem. Finally, we give some examples of H-Berwald and H-Landsberg metrics and an example of H-Landsberg metric which is not H-Berwaldian.