619 research outputs found
Looking for K\"ahler- 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 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.Comment: This article is in 15 page. arXiv admin note: text overlap with this
http URL and arXiv:1202.6202 by other autho
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