Looking for Kähler- Einstein Structure on Cartan Spaces with Berwald connection

A Cartan manifold is a smooth manifold M whose slit cotangent bundle T*M0 is endowed with a regular Hamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo)-Riemannian metric gij in the vertical bundle over T*M0 and using it a Sasaki type metric on T*M0 is constructed. A natural almost complex structure is also defined by K on T*M0 in such a way that pairing it with the Sasaki type metric an almost Kähler structure is obtained. In this paper we deform gij to a pseudo-Riemannian metric Gij and we define a corresponding almost complex Kähler 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*M0, G, J) is Kähler- Einstein, then the Cartan structure given by K reduce to a Riemannian one.

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