A global approach in real and complex Finsler geometry by averaging methods

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540086
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kagoshima University
• Principal Investigator: AIKOU Tadashi 鹿児島大学, 理工学研究科, 教授 (00192831)
• Co-Investigator(Kenkyū-buntansha): OBITSU Kunio 鹿児島大学, 大学院理工学研究科, 准教授 (00325763) ; MIYAJIMA Kimio 鹿児島大学, 大学院理工学研究科, 教授 (40107850)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: Finsler manifolds / Rizza structures / Rizza-negativity / Averaged metrics / Averaged connections

Outline of Final Research Achievements

Finsler geometry is the differential geometry of smooth family Hessian manifolds or Kahlerian manifolds parameterized by base points corresponding to real or complex category. In this research, we have investigated real or complex Finsler geometry by using the averaging method, namely, we consider the averaged metrics and averaged connection obtained by the integral along the fibers. In particular, we have investigated the conformal geometry in real category, and we have obtained a characterization of conformal flatness of Finsler metrics. In complex category, we have introduced the notion of Rizza-negativity of holomorphic vector bundles, and further, we investigated the ampleness or negativity of holomorphic vector bundles in terms of the curvature of Rizza-structure which is naturally defined in the tautological line bundles.

Free Research Field

幾何学