In this paper, we study Finsler gradient almost Ricci solitons of spherically symmetric type. We find the PDE characterization for a spherically symmetric Finsler manifold with spherically symmetric volume form to be a Finsler gradient almost Ricci soliton. As an application, we classify all polynomial spherically symmetric Finsler gradient almost Ricci solitons. These solitons must be steady or Douglas. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, a new notion of generalized super-weak Funk functions on a Finsler (or spray) space is introduced. We show that there is no non-trivial generalized super-weak Funk function on a complete spray space generalizing a result previously known only in the case of weak Funk functions on compact spray spaces. As its application, we prove that every complete Finsler space with S≥λτ ^2 for some positive constant λ must be Riemannian where S and τ are the S-curvature and the distortion respectively. We also give new or simple proofs of the Shen-Sun’s and Chen-Shen’s global rigidity theorems.

In this paper, we study a class of projective flat Finsler spaces with constant flag curvature and cohomogeneity not exceeding two. We find equations that characterize these metrics generalizing results previously only known in the case of spherically symmetric Finsler metrics. Moreover, we manufacture new 2-dimensional family of projectively flat Finsler metrics of negative constant flag curvature. These metrics contain Chern-Shen's construction in Riemann-Finsler geometry, Nankai Tracts in Mathematics, Vol 6 (World Scientific Publishing, Hackensack, NJ, 2005), X+192pp.

By deriving Reilly type inequality we prove a lower bound estimate of the first Dirichlet eigenvalue of the Finsler Laplacian operator on a compact Finsler measure space with smooth boundary (M,F,dμ ) and weighted Ricci curvature bound from below Ric _N≥ (N-1)k>0 , whose boundary has non-positive dμ -mean curvature. Moreover, we show that the lower bound is achieved if and only if M is isometric to a forward (backward) geodesic ball of radius π/2√(k) and F has constant radial flag curvature equal to k.

In this paper,we study a class of Finsler metrics of cohomogeneity two on R×R~n.They are called weakly orthogonally invariant Finsler metrics.These metrics not only contain spherically symmetric Finsler metrics and Marcal-Shen's warped product metrics but also partly contain another "warping" introduced by Chen-Shen-Zhao.We obtain differential equations that characterize weakly orthogonally invariant Finsler metrics with vanishing Douglas curvature,and therefore we provide a unifying frame work for Douglas equations due to Liu-Mo,Mo-Solórzano-Tenenblat and Solórzano.As an application,we obtain a lot of new examples of weakly orthogonally invariant Douglas metrics.

In this paper, we study a class of Finsler metrics of cohomogeneity not exceeding 2 on ℝ×ℝ^n . These metrics not only contain Finsler warped product metrics introduced by Marçal and Shen, but also contain the normalized Einstein universe in general relativity. We obtain the PDE characterization for these metrics to be Einstein refining the result of Marçal and Shen. Infinitely many new non-Riemannian Ricci-flat Finsler metrics of this type are constructed.

In this paper, we study an important Finslerian quantity, namely, principal curvatures. By establishing an explicit expression of principal curvatures of a Finsler metric with orthogonal invariance, we show that such Finsler metrics have at most two distinct principal curvatures in all directions. Furthermore one of these principal curvatures is simple when such Finsler metrics have two distinct principal curvatures. As an application, we completely determine all principal curvatures of a two-parameter family of Finsler metrics. We show that for these metrics, the simple principal curvature is simpler than another principal curvature.

In this paper, we give a lot of new non-projectively flat Finsler warped product metrics with zero flag curvature. We also manufacture new generalized Douglas–Weyl (GDW) metrics which are nontrivial in the sense that these metrics are neither Douglas metrics nor Weyl metrics via hyperbolic spaces.

Projectively flat Finlser metrics on a convex domain U in ℝ^n are regular solutions to Hilbert’s Fourth Problem. In this paper, we study projectively flat Finlser metrics on U. We find equations that characterize these metrics with weakly orthogonal invariance, refining a theorem due to Sol ó rzano-Le ó n. As its application, we obtain infinitely many new projectively flat Finlser metrics on 𝕊^n+1 and determine their scalar flag curvature. These metrics contain Bryant’s projective spherically symmetric Finsler metric of constant flag curvature 1.

In this paper, we study Ricci-quadratic sprays (or Finsler) spaces which are non-trivial in the sense that these sprays (or Finsler) spaces are not strongly Ricci-quadratic. First, we find infinitely many such sprays on an open domain in ℝ^n which are not induced by (not necessary positive definite) Finsler metrics. Then, we explicitly construct a lot of non-trivial Ricci-quadratic Finsler metrics by finding the PDE characterization for the spherically symmetric Finsler metrics to be Ricci-quadratic and strongly Ricci-quadratic.

In this paper, we give an estimate of the upper bound of the inner radius for a complete Finsler measure space (M,F,dμ ) of weighted Ricci curvature bounded from below, in terms of the dμ -mean curvature bound of the boundary. We also prove the rigidity result that when the boundary is compact, the upper bound is achieved if and only if M is isometric to a forward Finsler geodesic ball of vanishing radical flag curvature.

The Weyl curvature is one of the most fundamental quantities in projective Finsler geometry. In this paper, we study a class of Finsler warped product metrics with quadratic Weyl curvature. We give necessary and sufficient conditions of such metrics to be of quadratic Weyl curvature which are non-trivial in the sense that these metrics are not of Weyl type, refining a theorem due to Gabrani-Sevim-Shen. As its application, we construct infinitely many new non-trivial W-quadratic Finsler warped product metrics. In particular, we find non-trivial W-quadratic Finsler metrics which are not Douglas type.

In this paper, we study a class of Finsler metrics defined by a vector field on a gradient Ricci soliton. We obtain a necessary and sufficient condition for these Finsler metrics on a compact gradient Ricci soliton to be of isotropic S-curvature by establishing a new integral inequality. Then we determine the Ricci curvature of navigation Finsler metrics of isotropic S-curvature on a gradient Ricci soliton generalizing result only known in the case when such soliton is of Einstein type. As its application, we obtain the Ricci curvature of all navigation Finsler metrics of isotropic S-curvature on Gaussian shrinking soliton.

In this paper, we study a class of Finsler measure spaces whose weighted Ricci curvature satisfies R i c ∞ = c F 2 {\mathbf {Ric}}_{\infty }=cF^{2} . This class contains all gradient Ricci solitons and Finsler Gaussian shrinking solitons. Thus Finsler measure spaces in this class are called Finsler gradient Ricci solitons. For a Randers measure space, we find sufficient and necessary conditions for this space to be a Finsler gradient Ricci soliton. In particular, we show that Randers-Finsler gradient Ricci solitons must have isotropic S S -curvature. Finally, we give an equivalent condition for a Randers measure space to be a Finsler gradient Ricci soliton of constant S S -curvature.

In this paper, we study a class of Finsler metrics, called generalized Douglas–Weyl (GDW) metrics. By finding equations that characterize these metrics by warped product, we manufacture GDW-metrics which are non-trivial in the sense that these metrics are neither Douglas metrics nor Weyl metrics.

In this paper, we study a new Finslerian quantity T̂ defined by the T -curvature and the angular metric tensor. We show that the T̂ -curvature not only gives a measure of the failure of a Finsler metric to be of scalar flag curvature but also has a vanishing trace. We find that the T̂ -curvature is closely related to the Riemann curvature, the Matsumoto torsion and the Θ-curvature. We solve Z. Shen’s open problem in terms of the T̂ -curvature. Finally, we give a global rigidity result for Finsler metrics of negative Ricci curvature on a compact manifold via the T̂ -curvature, generalizing a theorem previously only known in the case of negatively curved Finsler metrics of scalar flag curvature.

We give an explicit construction of an (n+1)-parameter family of pro-jectively Ricci-flat Finsler metrics on an n-dimensional double hemisphere by using the spherical harmonics of degree 1. We also show that all locally projectively flat Ran-ders metrics with isotropic S-curvature on the standard unit sphere are projectively Ricci-flat.

The S-curvature is one of most important non-Riemannian quantities in Finsler geometry. In this paper, we study Finsler warped product metrics with isotropic S-curvature. We find the equation that characterizes Finsler warped product metrics of isotropic S-curvature. Then we explicitly construct infinitely many new non-spherically symmetric such metrics.

The weight Ricci curvature plays an important role in studying global Finsler geometry. In this paper, we study a class of Finsler measure spaces of constant weighted Ricci curvature. We explicitly construct new families of such complete Finsler measure spaces. In particular, we find an eigenfunction and its eigenvalue for such spaces, generalizing a result previously only known in the case of Gaussian shrinking soliton. Finally, we give necessary and sufficient conditions on the coordinate functions for these spaces to be Euclidean measure spaces.

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.