一类对偶平坦的黎曼度量

给出了黎曼度量局部对偶平坦的一个充分条件:黎曼度量的Spray所满足的方程。同时,指出该条件是非必要的,并给出了相关反例。进一步,对满足条件的这类黎曼度量的性质进行了研究。具体地,讨论了这类度量成为Einstein度量的条件。从黎曼曲率着手,通过计算发现:当空间维数n3,这类黎曼度量是Einstein度量,当且仅当它是欧氏度量;但是,这个结论对n=2的情形不适用。

A Class of Dually Flat Riemannian Metrics

In this paper, a sufficient condition of locally dual flat in Riemannian space is obtained：an equation that the spray of a Riemannian metric satisfies. At the same time, the theory what this condition is not necessary is pointed out since an example is given to prove. Further research is finished to characterize the quality of this kind of Riemannian metrics. The equivalent condition that this kind of locally dually flat Riemannian metric is Einstein metrics is disussed. The quality of this kind of locally dually flat Riemannian metric is been researched to show that they are Einstein metrics. Here Riemannian curvature is main consideration. A series of computation shows that a locally dually flat Riemannian metric is Einstein metric if and only if it is Euclidian with dimen-sion n≥3 . But this is not suitable for the space with dimension n=2.