On warped product Finsler spaces of Landsberg type

Using the Cartan connection for the study of the warped product Finsler spaces, we find the necessary and sufficient conditions for such manifolds to be Riemannian, Landsberg, Berwald, and locally Minkowski, separately. Using the theorems, we deal with the problem of finding non-Berwald Landsberg manifolds. Remarkably, we prove that a manifold of this type having the lowest dimension cannot be of warped product type. We also construct infinitely many new examples of Berwald manifolds of dimension 2 + k for every positive integer k, which are not locally Minkowski. We give the method of constructing infinitely many of such manifolds. As corollaries, we find the necessary and sufficient conditions for the Sasaki-Finsler metric G of a warped product Finsler manifold to be bundlelike for the vertical foliation, and the necessary and sufficient conditions on a warped product Finsler manifold to have the vertical foliation on the Riemannian manifold (M', G) be totally geodesic. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3638036]