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On Projectively Flat $(\alpha,\beta)$-metrics |
Canad. Math. Bull. 52(2009), 132-144
Published:2009-03-01
Printed: Mar 2009
Printed: Mar 2009
- Zhongmin Shen
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] |