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 = F-1 x(f) F-2 is of constant curvature c if and only if the base space F-1 is also of constant curvature c, the fiber space F-2 is of some constant curvature alpha, 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 F-1 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-Carton 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. (c) 2012 Elsevier B.V. All rights reserved.