项目来源

浙江省自然科学基金

项目主持人

李本伶

项目受资助机构

宁波大学

立项年度

2013

立项时间

未公开

项目编号

Y13A010070

研究期限

未知 / 未知

项目级别

省级

受资助金额

6.00万元

学科

数理科学部

学科代码

未公开

基金类别

一般项目

关键词

Finsler 度量 ; 旗曲率 ; 射影平坦 ; Ricci 曲率 ; 对偶平坦 ; Finsler metric ; flag curvature ; projectively flat ; Ricci curvature ; dually flat

参与者

未公开

参与机构

未公开

项目标书摘要：Finsler几何是微分几何中重要的研究领域。本研究围绕着Finsler 几何中的测地线，黎曼曲率,Ricci曲率等展开了深入的研究。本研究组对于研究范围内不同的几何问题重新构建相应的偏微分方程理论，在射影平坦具有常曲率的度量分类问题极其应用,Ricci 曲率的研究，以及对偶平坦度量的研究中得到了多个原创性成果。已发表的主要研究结果包括:1射影平坦具有常数旗曲率的Finsler 度量进行了分类刻画。在黎曼几何中射影平坦与具有常数截面曲率是等价的。而在更一般的度量空间中这一结果不再成立。由于此问题不仅是几何问题还涉及到复杂的非线性PDE,故长期以来未被解决。研究组对此问题进行了多年长期深入地研究，最终得到了此类度量的刻画分类。2Finsler 几何中一类新的Ricci 曲率张量的研究。在Finsler 几何以及Spray 几何中已有几个Ricci 曲率张量，比如用Ricci曲率定义的Hessian 等。研究组利用新的方法，引入了一个新的曲率张量并讨论了它与Ricci 曲率的关系以及与其他非Riemann 量的关系。3近年来，对偶平坦Finsler 度量的研究对信息几何的影响愈来愈深刻。我们在对对偶平坦度量的研究中发现其等价方程可以利用Hamel’s 方程来进行求解，故而对其进行了刻画并给出了具体构造的方法，得到了一大批非平凡的实例。4一类带有双重根号的Finsler 度量的研究。由于Finsler 度量计算的复杂性，构造特殊旗曲率的度量变得尤为重要。它也有助于对常曲率Finsler 度量的理解。本研究组对此类带有双重根号的射影平坦常旗曲率的Finsler 度量进行了分类刻画。

Application Abstract: Finsler geometry is an important part in differential geometry.Our research focuses on the geodesics,Riemann curvature,Ricci curvature and etc.We reconstruct the corresponding PDE theory to the geometric problems within our research field.Based on this,we study the classification of projectively flat Finsler metrics with constant flag curvature,the Ricci curvature and dually flat Finsler metrics,obtaining original achievements.With collaborators,we obtain the following results:1)The classification of projectively flat Finsler metrics with constant flag curvature.In Riemann geometry,projectively flat Riemann metrics are equivalent to metrics with constant section curvature.However,it is no longer true in general case.This is a problem not having beeen solved for quite long,for it is not only a geometric problem but also a PDE problem.We spend several years in this problem and find a way to work it out.Then we give the classification.2)The research on a new Ricci Tensor in Finsler geometry.There are several Ricci tensors in Finsler geometry and Spray geometry.By some new method,we introduce a new Ricci tensor to Finsler geometry.The relationship of this Ricci tensor and other non-Riemannian quantities are discovered.3)Recently,the research on dually flat metrics deeply influence information geometry.We find that the equivalent equation of dually flat metrics can be solved by Hamels equation.Then we characterize these metrics and obtain a group of non-trivial dually flat metrics.4)Finsler metrics with double square roots are studied.How to construct more Finsler metrics with special curvature is an important problem because it is difficult to do computation in Finsler geometry.This is also helpful to understand Finsler geometry a step further.We give the classification of these projectively flat Finsler metrics(with double square roots)with constant flag curvature.

项目受资助省

浙江省