芬斯勒几何中的Landsberg曲率及相关问题研究

发布时间：2018-07-21 10:44

【摘要】：本文主要围绕芬斯勒几何中一类重要的几何量——Landsberg曲率展开了深入研究。首先,我们对射影平坦的(α,β)-度量展开了研究,并分类刻画了射影平坦的Berwald(α,β)-度量和射影平坦的弱Landsberg(α,β)-度量以及射影平坦且具有相对迷向平均Landsberg曲率的(α,β)-度量。其次,我们研究了具有相对迷向平均Landsberg曲率(即J+c(x)FI=0)的闭的芬斯勒流形,并证明了若c(x)在流形上恒为正或恒为负,则该流形一定是黎曼流形。此外,本文研究了具有迷向平均Berwald曲率的芬斯勒度量F,并给出了F具有殆迷向S-曲率的一个充分条件。最后,与他人合作,本文给出了具有弱迷向旗曲率的芬斯勒度量所满足的一个偏微分方程组;还给出了具有标量旗曲率且具有常数平均Berwald曲率的芬斯勒度量的旗曲率K所满足的一个恒等式。
[Abstract]:This paper focuses on a class of important geometric quantities in Finsler geometry, Landsberg curvature. Firstly, we study the projective flat (伪, 尾) -metric, and characterize the projective flat Berwald (伪, 尾) -metric and the projectively flat weakly Landsberg (伪, 尾) -metric and the projective flat (伪, 尾) -metric with the relative isotropic average Landsberg curvature. Secondly, we study the closed Fensler manifold with the relative isotropic mean Landsberg curvature (that is, J c (x) FI0), and prove that if c (x) is always positive or negative on the manifold, then the manifold must be Riemannian manifold. In addition, we study the Fensler metric F with isotropic mean Berwald curvature, and give a sufficient condition for F to have almost isotropic S-curvature. Finally, in cooperation with others, we give a partial differential equation system of Fensler metric with weak isotropic flag curvature. An identity of the flag curvature K of the Fensler metric with scalar flag curvature and constant average Berwald curvature is also given.
【学位授予单位】：重庆理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.1

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