Projectively flat Asanov Finsler metric (Q2644626)

Let \(\alpha\) be a Riemannian metric and \(\beta\) some \(1\)-form on a smooth manifold \(M\). An \((\alpha,\beta)\)-metric is a Finsler metric of the type \(F=\alpha \phi(\beta/\alpha)\), where \(\phi\) is a positive smooth function satisfying some additional conditions. The metric is called Asanov-Finsler metric if \(\phi\) is of the form \[ \phi(s)=(s^2+g s+1)^\frac{1}{2} \exp\{G/2 \arctan[\beta/(h\alpha)+G/2]\} \] with \(g \in (-2,2), h=(1-g^2/4)^\frac{1}{2}, G=g/h\). The conditions under which an Asanov-Finsler metric is locally projectively flat are derived using Hamels criterion. The main theorem states that this is the case if and only if \(\alpha\) is locally projectively flat and \(\beta\) is parallel with respect to \(\alpha\). It is also shown that the second condition is equivalent to the vanishing of the Douglas tensor of \(F\).