n the present paper, we give an answer to a question which is closely related to
doubly warped product of Finsler metrics: “For each n, is there an n-dimensional
Finsler manifold (M, F ), admitting a non-constant smooth function f on M such
that ∂f
∂xi
∂gij
∂yk = 0?”. We relate the preceding mentioned condition to different
concepts appeared and studied in Finsler geometry. We introduce and investigate
the notion of a semi concurrent vector field on a Finsler manifold. We show that some
special Finsler manifolds admitting such vector fields turn out to be Riemannian. We
prove that Tachibana’s characterization of Finsler manifolds admitting a concurrent
vector field leads to Riemannian metrics. Various examples for conic Finsler spaces
that admit semi-concurrent vector field are presented.