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20 April 2024
 
  » arxiv » math.DG/0109037

 Article overview


Funk Metrics and R-Flat Sprays
Zhongmin Shen ;
Date 5 Sep 2001
Subject Differential Geometry; Metric Geometry MSC-class: 53B60 | math.DG math.MG
AbstractThe well-known Funk metric F(x,y) is projectively flat with constant flag curvature K=-1/4 and the Hilbert metric H(x,y):=(F(x,y)+F(x,-y))/2 is projectively flat with constant curvature K=-1. These metrics are the special solutions to Hilbert’s Fourth Problem. In this paper, we construct a non-trivial R-flat spray using the Funk metric. It is then an inverse problem in the calculus of variation to find a Finsler metric that induces the R-flat spray. We find an explicit solution to this inverse problem and obtain a non-trivial projectively flat Finsler metric with K=0.
Source arXiv, math.DG/0109037
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