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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.

In this paper, we introduce the notion of weakly symmetric Finsler spaces and study some geometrical properties of such spaces. In particular, we prove that each maximal geodesic in a weakly symmetric Finsler space is the orbit of a one-parameter subgroup of the full isometric group. This implies that each weakly symmetric Finsler space has vanishing S-curvature. As an application of these results, we prove that there exist reversible non-Berwaldian Finsler metrics on the 3-dimensional sphere with vanishing S-curvature. This solves an open problem raised by Z. Shen.

The collection of all projective vector fields on a Finsler space (M, F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra. A specific Lie sub-algebra of projective algebra of Randers spaces (called the special projective algebra) of non-zero constant S-curvature is studied and it is proved that its dimension is at most . Moreover, a local characterization of Randers spaces whose special projective algebra has maximum dimension is established. The results uncover somehow the complexity of projective Finsler geometry versus Riemannian geometry.

The Riemann curvature in Riemann–Finsler geometry can be regarded as a collection of linear operators on the tangent spaces. The algebraic properties of these operators may be linked to the geometry and the topology of the underlying space. The principal curvatures of a Finsler space (M, F) at a point x are the eigenvalues of the Riemann curvature operator at x. They are real functions κ on the slit tangent manifold TM₀. A principal curvature κ(x, y) is said to be isotropic (respectively, quadratic) if κ(x, y)/F(x, y) is a function of x only (respectively, κ(x, y) is quadratic with respect to y). On the other hand, the Randers metrics are the most popular and prominent metrics in pure and applied disciplines. Here, it is proved that if a Randers metric admits an isotropic principal curvature, then F is of isotropic S-curvature. The same result is also established for F to admit a quadratic principal curvature. These results extend Shen's verbal results about Randers metrics of scalar flag curvature K = K(x) as well as those Randers metrics with quadratic Riemann curvature operator. The Riemann curvature may be broken into two operators and . The isotropic and quadratic principal curvature are characterized in terms of the eigenvalues of and .

A Finsler metric F is said to be spherically symmetric if the orthogonal group O(n) acts as isometries of F. In this paper, we show that every spherically symmetric Finsler metric of isotropic Berwald curvature is a Randers metric. We also construct explicitly a lot of new isotropic Berwald spherically symmetric Finsler metrics.

In the present paper, we prove that a general $(\alpha ,\beta )$-metric $F$ is of isotropic $S$-curvature if and only if it is of isotropic $E$-curvature under an extra condition on $\alpha$ and $\beta$.

The study of curvature properties of homogeneous Finsler spaces with $(\alpha ,\beta )$-metrics is one of the central problems in Riemann–Finsler geometry. In this paper, the existence of invariant vector fields on a homogeneous Finsler space with square metric is proved. Further, an explicit formula for $S$-curvature of a homogeneous Finsler space with square metric is established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of aforesaid $(\alpha ,\beta )$-metric is calculated.