The study of Bott connection and its applications to Finsler geometry

Bott连接的研究及其在芬斯勒几何中的应用

• 批准号: 13640084
• 负责人: AIKOU Tadashi
• 金额: $ 2.11万
• 依托单位: Kagoshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640084/
• 关键词: Finsler metrics; Bott connections; Kahler fibrations; Projective flatness; 複素Finsler計量; 複素Finsler接続; 擬ケーラー計量

In this research, we have investigated complex Bott connections and its applications to complex Finsler geometry in the period 2001-2002. In 2001, we have mainly studied some relations between the complex Bott connections and complex Finsler connections. Under the results obtained in 2001, we have studied the flatness and projective flatness of complex Finsler metrics, and moreover, the differential geometry of Kahler fibrations with pseudo Kahler metrics in 2002.The main contents of this research is the investigation of complex Finsler connection which is introduced on the relative tangent bundle over the projective bundle. In terms of this connection, we can investigate the projective flatness of complex Finsler metrics, and finally we obtained the projective curvature which is the obstruction of projective flatness. Such an investigation leads us naturally to the study of minimal ruled surface over a compact Riemann surface. In fact, the Kahler metric on such a surface induces a complex Finsler metric with negative curvature in the corresponding vector bundle. The basic and important fact is that the projective flatness of the corresponding Finsler metric is equivalent to that the minimal ruled surface is the total space of a Kahler submersion with isometric fibers to the base Riemann surface.The main results in this research are contained in "Kahler fibrations and complex Finsler geometry (preprint, 2002)" and "A note on some special Finsler manifold (preprint, 2002)".

在这项研究中，我们研究了2001-2002年间复杂的Bott连接及其在复杂Finsler几何中的应用。2001年，我们主要研究了复Bott联络和复Finsler联络之间的一些关系。在2001年的结果下，我们研究了复Finsler度量的平坦性和射影平坦性，并在2002年研究了具有伪Kahler度量的Kahler原函数的微分几何。本研究的主要内容是研究射影丛上相对切丛上的复Finsler联络。在这种联系下，我们可以研究复Finsler度量的射影平坦性，最终得到射影平坦度的射影曲率。这样的研究自然将我们引向紧致黎曼曲面上的极小直纹曲面的研究。事实上，这种曲面上的Kahler度量在相应的向量丛中诱导出一个负曲率的复Finsler度量。基本的和重要的事实是，对应的Finsler度量的射影平坦性等价于极小直纹面是具有到基本Riemann曲面的等距纤维的Kahler浸没的全空间。本研究的主要结果包含在《Kahler纤毛和复杂Finsler几何(Preprint，2002)》和《关于某些特殊Finsler流形的注记(Preprint，2002)》中。

项目成果

Kimio Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities, in Complex Analysis and Related Topics, Proceedings of the Japan-Korea Joint Workshop in Mathematics 2001"115-124 (2002)

Kimio Miyajima：“复杂分析和相关主题中正常孤立奇点边界上 CR 结构的变形理论，2001 年日韩数学联合研讨会论文集”115-124 (2002)

K.Miyajima: "Deformation theory of CR structures on a boundary of normal isolated singularities"Complex Analysis and Related Topics, Proceedings of the Japan-Korea Joint Workshop in Mathematics 2001. 115-124 (2002)

K.Miyajima：“正常孤立奇点边界上 CR 结构的变形理论”复分析及相关主题，日韩数学联合研讨会论文集 2001. 115-124 (2002)

Tadashi Aikou: "Applications of Bott connection to Finsler geometry"Steps in Differential Geometry, The Institute of Mathematics and Informatics, University of Debrecen. 1-13 (2001)

Tadashi Aikou：“Bott 连接在芬斯勒几何中的应用”微分几何的步骤，德布勒森大学数学与信息学研究所。

Tadashi Aikou: "Differential geometry of K{\"a}hler fibrations and its application to Finsler geometry"Far East Journal of Mathematical Science. 4. 91-117 (2002)

Tadashi Aikou：“K{”a}hler 纤维的微分几何及其在 Finsler 几何中的应用”远东数学科学杂志. 4. 91-117 (2002)

Tadashi Aikou: "Differential geometry of Kaher fibrations and its application to Finsler geometry"Far East Journal of Mathematical Science. No. 4. 91-117 (2002)

Tadashi Aikou：“卡赫纤维的微分几何及其在芬斯勒几何中的应用”远东数学科学杂志。