具有某种特定属性的芬斯勒度量的构造

发布时间：2018-08-06 17:13

【摘要】：芬斯勒几何是没有二次型限制的黎曼几何,其理论和研究方法在信息科学和计算机技术等方面有着广泛的应用,成为21世纪微分几何的发展方向.构造对偶平坦的芬斯勒度量是芬斯勒几何中一类极具研究价值的问题.本文基于这一点通过求解对偶平坦偏微分方程,给出了一些对偶平坦芬斯勒度量新的例子.全文主要分为两个部分:第一部分:研究在欧氏空间的某个凸集Ω上的球对称芬斯勒度量F(x,y)=|y|(?),通过求解该芬斯勒度量的对偶平坦方程sfts + fss - 2ft=0,其中t=|x|2/2,s=x,y/|y|.给出了两类对偶平坦球对称芬斯勒度量.其中一类是函数f是多项式形式,另一类是应用幂级数的形式求解该方程.第二部分:研究广义(α,β)度量F = αφ(b2,β/α),其中α是黎曼度量,β是1形式,b = ||β||α,其对偶平坦方程为(φ2)2+φφ22 + 2sφ1φ2+2sφφ12 -4φφ1=0,通过引入变量ψ =φ给出对偶平坦另一形式的等价偏微分方程φ2 + 2sφ12 - 4φ1=0,求解此方程给出了多项式形式及变量分离形式的特解进而得到新的广义(α,β)度量是对偶平坦的芬斯勒度量.
[Abstract]:Finsler geometry is a Riemann geometry with no quadratic constraints. Its theory and research methods have been widely used in information science and computer technology and become the development direction of differential geometry in the 21st century. The construction of dual flat Fensler metric is an extremely valuable problem in Finsler geometry. Based on this, some new examples of dual flat Finsler metric are given by solving dual flat partial differential equations. This paper is mainly divided into two parts: in the first part, we study the spherically symmetric Finsler metric F (XFN y) = Gy (?) on a convex set 惟 of Euclidean space. By solving the dual flat equation sfts fss 2ftg 0 of the Fensler metric, where t = x 2 / 2 / 2% SXMY / AII. Two classes of dual flat spherically symmetric Finsler metrics are given. One is that the function f is polynomial and the other is solving the equation by using the form of power series. In the second part, we study the generalized (伪, 尾) metric F = 伪蠁 (b2, 尾 / 伪), where 伪 is Riemannian metric, 尾 is a form B = 尾伪, and the dual flat equation is (蠁 2) 2 蠁 22 2s 蠁 1 蠁 22 s 蠁 1 蠁 2 2s 蠁 12 -4 蠁 10 0. By introducing the variable 蠄 = 蠁, we give the equivalent partial differential of dual flatness. The equation 蠁 22 s 蠁 12 -4 蠁 1 0 is solved. The special solutions of polynomial form and variable separation form are given, and the new generalized (伪, 尾) metric is a dual flat Finsler metric.
【学位授予单位】：安徽师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O186.1

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