The geometry of geodesics and its application to the discrete mathematics

2014 Fiscal Year Final Research Report

• Project/Area Number: 22540072
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Niigata University
• Principal Investigator: INNAMI Nobuhiro 新潟大学, 自然科学系, 教授 (20160145)
• Project Period (FY): 2010-04-01 – 2015-03-31
• Keywords: 測地線 / 曲面 / ボロノイ図 / カットローカス / シュタイナー比 / 平面凸ビリヤード問題 / トポノゴフの比較定理 / 球面定理

Outline of Final Research Achievements

In an intrinsic metric space, a locally minimizing curve is called a geodesic. We say that a metric space is a geodesic space if any two points can be joined by a minimizing geodesic. H. Busemann put forward the geometry of geodesics in 1955 to study some properties of geodesics, the topological and metric structure of spaces. Using his methods, we produced results on the studies of sets of poles in a Riemannian manifold, the geodesic flows on surfaces and the plane convex billiard ball problems, the Steiner ratio problem for surfaces, a generalization of Toponogov's comparison theorem and some sphere theorems, the relation of a Voronoi diagram and the cut locus. We have a prospect to develop the geometry of geodesics in a non-symmetric intrinsic distance space.

Free Research Field

幾何学