The Study On Some Important Classes Of Locally Dually Flat (α,β)-metrics

In this paper, we mainly study some important classes of locally dually flat (α,β)-metrics, whereαdenotes a Riemannian metric on a manifold andβdenotes a 1-form.We first characterize locally dually flat Randers metric under the condition thatαis locally projectively flat. Further, we characterize locally dually flat (α,β)-metric in the form F = (α+β)~2/αunder certain conditions aboutαandβ. We also find some equations that characterize locally dually flat Matsumoto metric F =α~2/(α-β)and classify those with isotropic S -curvature. Finally, we characterize two important classes of locally projectively flat (α,β)-metrics of isotropic S -curvature.