Locally dually flat Finsler metrics with special curvature properties

Abstract

Locally dually flat Finsler metrics are studied in Finsler information geometry and naturally arise from the investigation of the so-called flat information structure. In this survey article, we first characterize locally dually flat and projectively flat Finsler metrics. Then we mainly study locally dually flat Randers metrics in the form F = α + β , where α is a Riemannian metric and β is a 1-form on the manifold. We find some equations that characterize locally dually flat Randers metrics and classify locally dually flat Randers metrics with weak isotropic flag curvature. Further, we characterize locally dually flay ( α , β ) -metrics in the form F = α + ϵ β + k β 2 α of scalar flag curvature, where ϵ, k are nonzero constants.