On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki–Finsler metrics

As an opening, we prove that a warped product Finsler space F=F1×fF2F=F1×fF2 is of constant curvature cc if and only if the base space F1F1 is also of constant curvature cc, the fiber space F2F2 is of some constant curvature αα, and five other partial differential equations are satisfied. A rather similar result is proved for the case of warped product Finsler spaces of scalar curvature. Close relationships between the geometry of the warped product Finsler spaces of constant curvature and the spectral theory of the Laplacian (Laplace–Beltrami operator) of the well-known Sasaki–Finsler metrics of the base space F1F1 is established by detailed investigation of the above mentioned PDEs. We also define a new tensor for warped product Finsler spaces, which we call a warped-Cartan tensor. Using the tensor we define a new class of warped product Finsler spaces, calling them C-Warped spaces, which contain Landsberg, Berwald, locally Minkowski and Riemannian spaces, but not necessarily all of the constant curvature Finsler spaces of warped product type. Several results are obtained and special cases, for example the case of Riemannian, C-Warped and projectively flat spaces are also considered.