arXiv subjects

Benling Li

Publications and source records attributed to Benling Li.

9 recordsLinked to original sources

arXiv2026

On Spherically Symmetric Sprays

This paper studies spherically symmetric sprays, i.e., sprays that are invariant under orthogonal transformations. We first establish a canonical form for such sprays, showing that their geodesic coefficients can be expressed as $G^i = |y|\alpha(r,s) y^i + |y|^2\beta(r,s) x^i$, where $r = |x|^2$ and $s = \langle x,y\rangle/|y|$. For projectively flat spherically symmetric sprays -- which are directly related to Hilbert's fourth problem on characterizing metrics whose geodesics are straight lines -- we derive a complete classification of those with isotropic curvature, and in particular, we obtain the explicit form of those with zero curvature. Furthermore, we characterize sprays of weakly isotropic curvature in this class by a system of partial differential equations. These results may provide a unified framework for understanding symmetry and curvature in spray geometry and could offer new insights into the metrizability problem in Finsler geometry.

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arXiv2026

Unexpected Analytic Phenomena on Finsler Manifolds

In the Riemannian setting, every flat Cartan--Hadamard manifold is isometric to Euclidean space, the canonical model that underlies the theory of Sobolev spaces and guarantees the sharpness/rigidity of the Hardy inequality, the uncertainty principle, and the Caffarelli--Kohn--Nirenberg (CKN) inequality. In this paper, we show that on a flat Finsler Cartan--Hadamard manifold -- Berwald's metric space -- the classical picture alters radically: the Nash embedding theorem fails, the Sobolev space becomes nonlinear, and the Hardy and uncertainty inequalities break down completely, whereas the CKN inequality exhibits a sharp threshold in its validity depending on a parameter. By contrast, on Funk metric spaces -- another class of Finsler Cartan--Hadamard manifolds -- this threshold behavior disappears, although all the other non-Riemannian features persist. We trace this divergence to the lower bound of the $S$-curvature. As a consequence, the failure of the aforementioned functional inequalities is established for a broad class of Finsler manifolds.

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arXiv2025

Hilbert's fourth problem in the constant curvature setting

Hilbert's fourth problem seeks the classification of metric geometries where straight lines are shortest paths. Its regular case identifies the projectively flat Finsler manifolds. This broader framework breaks the equivalence between projective flatness and constant curvature that holds in the Riemannian setting, creating a more intricate classification problem. This paper resolves the long-standing question of how the local structure determines the global topology for such manifolds of constant flag curvature, where flag curvature is the natural generalization of Riemannian sectional curvature. We derive explicit distance formulas for all cases of constant flag curvature. For non-positive constant curvature, we establish a global classification of forward complete manifolds, a uniqueness theorem for forward complete metrics, and a characterization of maximal domains of metrics where exotic examples are constructed. For positive constant curvature, we prove a maximum diameter theorem and show the completion of such manifold is a sphere. A fundamental connection is revealed between Sobolev space nonlinearity and backward incompleteness. This work provides a complete characterization of the global geometry for the regular case of Hilbert's fourth problem with constant flag curvature.

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arXiv2024

Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature

The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive $S$-curvature, cf. Huang, Krist\'aly and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the $S$-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the $S$-curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the $S$-curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the $S$-curvature plays again a decisive role.

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arXiv2017

Finsler metrics of weakly isotropic flag curvature

Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an $n$-dimensional manifold $M$ we study the Finsler metric $F=F(x,y)$ of scalar flag curvature ${\bf K} = {\bf K}(x,y)$ and discover some equations ${\bf K}$ should be satisfied. As an application, we mainly study the metric $F$ of weakly isotropic flag curvature ${\bf K} = \frac{3 θ}{F} + σ$, where $θ=θ_i(x) y^i \neq 0$ is a $1$-form and $σ=σ(x)$ is a scalar function. We prove that in this case, $F$ must be a Randers metric when $dim(M) \geq 3$. Further, without the restriction on the dimension we prove that projectively flat Finsler metrics of such weakly isotropic flag curvature are Randers metrics too.

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arXiv2017

On Landsberg general $(α,β)$-metrics with a conformal 1-form

In this paper, we study almost regular Landsberg general $(α,β)$-metrics in Finsler geometry. The corresponding equivalent equations are given. By solving the equations, we give the classification of Landsberg general $(α,β)$-metrics under the conditon that $β$ is closed and conformal to $α$. Under this condition, we prove that regular Landsberg general $(α,β)$-metrics must be Berwaldian when the dimension is greater than two. For the almost regular case, the classification also is given and some new non-Berwaldian Landsberg metrics are found.

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arXiv2016

On Douglas general $(α,β)$-metrics

Douglas metrics are metrics with vanishing Douglas curvature which is an important projective invariant in Finsler geometry. To find more Douglas metrics, in this paper we consider a class of Finsler metrics called general $(α,β)$-metrics, which are defined by a Riemannian metric $α=\sqrt{a_{ij}(x)y^iy^j}$ and a $1$-form $β=b_i(x)y^i$. We obtain the differential equations that characterizes these metrics with vanishing Douglas curvature. By solving the equivalent PDEs, the metrics in this class are totally determined. Then many new Douglas metrics are constructed.

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arXiv2015

On a class of projectively flat Finsler metrics

In this paper, we study a class of Finsler metrics composed by a Riemann metric $α=\sqrt{a_{ij}(x)y^i y^j}$ and a $1$-form $β=b_i(x)y^i$ called general ($α$, $β$)-metrics. We classify those projectively flat when $α$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of projectively flat Finsler metrics is found.

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arXiv2013

On the classification of projectively flat Finsler metrics with constant flag curvature

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way. Further, we construct a new projectively flat Finsler metric with flag curvature ${\bf K}=1$ determined by a Minkowskian norm with double square roots at the origin. As an application of our main theorems, we give the classification of locally projectively flat spherical symmetric Finsler metrics much easier than before. ----Comments are welcome.

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