Regarding the Kahler-Einstein structure on Cartan spaces with Berwald connection

A Cartan manifold is a smooth manifold M whose slit cotangent bundle T*M-0 is endowed with a regular Hamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo)-Riemannian metric g(ij) in the vertical bundle over T*M-0 and using it, a Sasaki type metric on T*M-0 is constructed. A natural almost complex structure is also defined by K on T*M-0 in such a way that pairing it with the Sasaki type metric an almost Kahler structure is obtained. In this paper we deform g(ij) to a pseudo-Riemannian metric G(ij) and we define a corresponding almost complex Kahler structure. We determine the Levi-Civita connection of G and compute all the components of its curvature. Then we prove that if the structure (T*M-0, G, J) is Kahler- Einstein, then the Caftan structure given by K reduces to a Riemannian one.