## 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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