Funk Metrics and R-Flat Sprays ∗
نویسنده
چکیده
The well-known Funk metric F (x, y) is projectively flat with constant flag curvature K = −1/4 and the Hilbert metric Fh(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.
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