在具有Berwald联络的Cartan空间上寻找Kähler-Einstein结构
Looking for Kähler- Einstein Structure on Cartan Spaces with Berwald connection
摘要：
一个Cartan流形是一个光滑流形M，它的有缝余切丛T*M0被赋予一个正则哈密顿量K，这个正则哈密顿量K是2次动量正齐次的。哈密顿量K定义了T*M0上垂直丛中的一个（伪）-黎曼度量gij，并利用它构造了T*M0上的一个Sasaki型度量.一个自然的几乎复结构也由K在T*M0上定义，使得将它与Sasaki型度量配对得到几乎Kähler结构。在本文中，我们变形gij伪黎曼度量Gij，我们定义了相应的几乎复Kähler结构。我们确定G的Levi-Civita联络并计算其曲率的所有分量。然后证明了如果结构（T*M0，G，J）是Kähler-Einstein结构，则由K给出的Cartan结构约化为Riemann结构.
英文摘要：
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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