On the covariant coefficients of geodesic sprays on Finsler manifolds.

For a Finsler metric F , we introduce the notion of F -covariant coefficients H i of the geodesic spray of F (Definition 3.1). We study some geometric consequences concerning the objects H i . If the F -covariant coefficients H i are written in the form H i = ∂ ̇ i H , for some smooth function H on M , 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 1 2 y i ∂ i F 2 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 n 3 functions H ij h resulting from the F -covariant coefficients do not form a linear connection. We find out that in the case of projectively flat metrics, the n 3 functions H ij h 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 H i of projectively 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. [ABSTRACT FROM AUTHOR]