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

Abstract

A Cartan manifold is a smooth manifold M whose slit cotangent bundle 0T *M is endowed with a regularHamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo)-Riemannian metric ij g in the vertical bundle over 0 T *M and using it, a Sasaki type metric on 0 T *M is constructed. A natural almost complex structure is also defined by K on 0 T *M in such a way that pairing it with the Sasaki type metric an almost Kähler structure is obtained. In this paper we deform ij g to a pseudo-Riemannian metric ij G 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 ( , , ) 0 T *M G J is Kähler- Einstein, then the Cartan structure given by K reduces to a Riemannianone.