R_1~3中类时曲面的局部Gauss-Bonnet公式

Local Gauss-Bonnet Formula for Timelike Surfaces in the Lorentzian Space R_1~3

研究洛伦兹空间R31中的类时曲面M上非退化曲线的测地曲率,得到计算测地曲率的Liouville公式,并利用Green公式把欧氏空间中经典的Gauss-Bonnet公式推广到R31中的类时曲面M的局部区域D上。证明了在D的边界曲线上,测地曲率的积分可能是发散的,因而此时的局部Gauss-Bonnet公式与欧氏空间中的Gauss-Bonnet公式有所不同。

In the present paper we study the geodesic curvature κg of non-degenerate curves on a timelike surface M,which is immersed in the Lorentzian space R31.We get a computation formula of Liouville type for geodesic curvature.By using the Green＇s formula we extend the Gauss-Bonnet formula to a local region D of M.It is proved that the integration of κg on the boundary of D may be divergent.Therefore the Gauss-Bonnet formula is different from the case of surface in the Euclidean space.