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On a class of projectively flat metrics with constant flag curvature. (English) Zbl 1157.53014

The authors determine the equations which characterize locally projectively flat Finsler metrics of the form \(F=(\alpha+\beta)^2/\alpha\), with \(\alpha=\sqrt{a_{ij}y^iy^j}\) and \(\beta=b_iy^i\) (where \(a_{ij}\) are the coefficients of a Riemannian metric and \(b_i\), the coefficients of a 1-form). Further, for the more general metrics \(F=\alpha+\varepsilon\beta+k\beta^2/\alpha\), \(\varepsilon\in\mathbb R\), \(k\in\mathbb R\backslash\{0\}\), the same goal is achieved, and the local structure is completely determined in the case of constant flag curvature.

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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