2001/09/05 by Zhongmin Shen, Shen, Zhongmin
Mathematics · #53B60 #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #math.DG #math.MG #msc:53B60
paper · pdf · doi:10.48550/arxiv.math/0109037
Revised in June, 2001, 10 pages, other related papers can be downloaded from http://www.math.iupui.edu/~zshen/
arxiv created 2001/09/05 · arxiv updated 2009/11/30
The 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.