Bulletin of the
Korean Mathematical Society BKMS

In this paper, we study a class of Finsler metric. First, we find some rigidity results of the dually flat $(\alpha,\beta)$-metric where the underline Riemannian metric $\alpha$ satisfies nonnegative curvature properties. We give a new geometric approach of the Monge-Amp\'ere type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.

Keywords: Finsler metric, $(\alpha,\beta)$-metric, dually flat, Monge-Amp\'ere equation, Bernstein type theorem

MSC numbers: 53B40, 53C60, 35B08