On Twisted Products Finsler Manifolds

On the product of two Finsler manifolds , we consider the twisted metric which is constructed by using Finsler metrics and on the manifolds and , respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study C-reducible and semi-C-reducible properties of this manifold. Then we obtain the Riemannian curvature and some of non-Riemannian curvatures of the twisted product Finsler manifold such as Berwald curvature, mean Berwald curvature, and we find the relations between these objects and their corresponding objects on and . Finally, we study locally dually flat twisted product Finsler manifold. 1. Introduction Twisted and warped product structures are widely used in geometry to construct new examples of semi-Riemannian manifolds with interesting curvature properties (see [1–3]). Twisted product metric tensors, as a generalization of warped product metric tensors, have also been useful in the study of several aspects of submanifold theory, namely, in hypersurfaces of complex space forms [4], in Lagrangian submanifolds [5], in decomposition of curvature netted hypersurfaces [6], and so forth. The notion of twisted product of Riemannian manifolds was mentioned first by Chen in [7] and was generalized for the pseudo-Riemannian case by Ponge and Reckziegel [8]. Chen extended the study of twisted product for CR-submanifolds in K？hler manifolds [9]. On the other hand, Finsler geometry is a natural extension of Riemannian geometry without the quadratic restriction. Therefore, it is natural to extend the construction of twisted product manifolds for Finsler geometry. In [10], Kozma-Peter-Shimada extended the construction of twisted product for the Finsler geometry. Let and be two Finsler manifolds with Finsler metrics and , respectively, and let be a smooth function. On the product manifold , we consider the metric for all and , where is the slit tangent manifold . The manifold endowed with this metric, we call the twisted product of the manifolds and and denote it by . The function will be called the twisted function. In particular, if is constant on , then is called warped product manifold. Let be a Finsler manifold. The second and third order derivatives of at are the symmetric trilinear forms and on , which called the fundamental tensor and Cartan torsion, respectively. A Finsler metric is called semi-C-reducible if its Cartan tensor is given by where and are scalar function on , is the angular metric, and [11]. If , then is called C-reducible Finsler metric, and if , then is called -like metric. The geodesic