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Locally dually flat Finsler metrics arise from information geometry. Such metrics have special geometric properties. In this paper, we characterize locally dually flat and projectively flat Finsler metrics and study a special class of Finsler metrics called Randers metrics which are expressed as the sum of a Riemannian metric and a one-form. We find some equations that characterize locally dually flat Randers metrics and classify those with isotropic S-curvature.

In this paper, we study the non-Riemannian quantity H in Finsler geometry. We obtain some rigidity theorems of a compact Finsler manifold under some conditions related to H. We also prove that the S-curvature for a Randers metric is almost isotropic if and only if H almost vanishes. In particular, S-curvature is isotropic if and only if H = 0.

In this paper, we introduce the notions of R-reversibility and Ricci-reversibility. We prove that Randers metrics are R-reversible or Ricci-reversible if and only if they are R-quadratic or Ricci-quadratic, respectively. Besides, we discuss the properties of Ricci- or R-reversible Randers metrics which are also weakly Einsteinian, or Douglassian, or of scalar flag curvature. In particular, we determine the local structure of Randers metrics which are Ricci-reversible and locally projectively flat, and prove that an n (≥ 3)-dimensional Ricci-reversible Randers metric of non-zero scalar flag curvature is locally projectively flat.

Locally dually flat Finsler metrics arise from information geometry. In this paper, we study locally dually flat Kropina metrics and find some equations that characterize locally dually flat Kropina metrics and classify those with scalar flag curvature. Finally, we also classify dually flat Kropina metrics with isotropic $S$-curvature.

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that $\beta$ is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

In this paper, we establish a closer relation between the Berwald scalar curvature and the $S$-curvature. In fact, we prove that a Finsler metric has isotropic Berwald scalar curvature if and only if it has weakly isotropic $S$-curvature. For Finsler metrics of scalar flag curvature and of weakly isotropic $S$-curvature, they have almost isotropic $S$-curvature if and only if the flag curvature is weakly isotropic.