关于Finsler曲面的几何和拓扑(英文)

Geometry and Topology on a Finsler Surface

本文在Ｆｉｎｓｌｅｒ曲面上定义了一个新的不变量Ｈ．该不变量等于零刻画了Ｒｉｅｍａｎｎ流形．文章给出了Ｈ的一个上界并且构造了Ｈ为常值的非Ｒｉｅｍａｎｎ的Ｆｉｎｓｌｅｒ曲面．此外，本文还推广了Ｌａｎｄｓｂｅｒｇ曲面的Ｇａｕｓｓ－Ｂｏｎｎｅｔ－Ｃｈｅｒｎ定理并分类了非正曲率的Ｆｉｎｓｌｅｒ曲面．

A new invariant H on a Finsler surface, which give a measure of the failure of Finsler surface to be Riemann, is defined. A upper bound of H is obtained. Non-Riemann Finsler surfaces with constant H are constructed. Gauss-Bonnet-Chern theorem for Landsberg surfaces is generalized. A classification of non-positive curved Finsler surfaces is given.