Funk Metrics and R-Flat Sprays
Differential Geometry
2007-05-23 v1 Metric Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.math/0109037,
title = {Funk Metrics and R-Flat Sprays},
author = {Zhongmin Shen},
journal= {arXiv preprint arXiv:math/0109037},
year = {2007}
}
Comments
Revised in June, 2001, 10 pages, other related papers can be downloaded from http://www.math.iupui.edu/~zshen/