  In this paper, we introduce horizontal and vertical warped product Finsler
manifold. We prove that every C-reducible or proper Berwaldian doubly warped
product Finsler manifold is Riemannian. Then, we find the relation between
Riemmanian curvatures of doubly warped product Finsler manifold and its
components, and consider the cases that this manifold is flat or has scalar
flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped
product manifolds and find a condition under which the horizontal and vertical
tangent bundles are totally geodesic. Also, we obtain some conditions under
which a foliated manifold reduces to a Reinhart manifold. Finally, we study an
almost complex structure on the slit tangent bundle of a doubly warped product
Finsler manifold.
