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# On Projectively Flat $(\alpha,\beta)$-metrics

Published:2009-03-01
Printed: Mar 2009
• Zhongmin Shen
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## Abstract

The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $(\alpha,\beta)$-metrics defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n \geq 3$.
 MSC Classifications: 53B40 - Finsler spaces and generalizations (areal metrics) 53C60 - Finsler spaces and generalizations (areal metrics) [See also 58B20]