Looking for K\"ahler- Einstein Structure on Cartan Spaces with Berwald connection

TL;DR

This paper investigates conditions under which a Cartan manifold's associated almost Kähler structure becomes Kähler-Einstein, concluding that such a structure implies the Cartan structure is Riemannian.

Contribution

It introduces a deformation of the metric on the cotangent bundle and characterizes when the resulting structure is Kähler-Einstein, linking it to Riemannian geometry.

Findings

Kähler-Einstein condition implies the Cartan structure is Riemannian

Explicit computation of Levi-Civita connection and curvature components

Deformation of the metric G leads to new insights into Cartan space structures

Abstract

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 K\"ahler structure is obtained. In this paper we deform gij to a pseudo-Riemannian metric Gij and we define a corresponding almost complex K\"ahler 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 K\"ahler- Einstein, then the Cartan structure given by K reduce to a Riemannian one.