Complete Finsler Manifold Research Articles

On the Cut Locus of Submanifolds of a Finsler Manifold

In this article, we investigate the cut locus of closed (not necessarily compact) submanifolds in a forward complete Finsler manifold. We explore the deformation and characterization of the cut locus, extending the results of Basu and Prasad (Algebr Geom Topol 23(9):4185–4233, 2023). Given a submanifold N, we consider an N-geodesic loop as an N-geodesic starting and ending in N, possibly at different points. This class of geodesics were studied by Omori (J Differ Geom 2:233–252, 1968). We obtain a generalization of Klingenberg’s lemma for closed geodesics (Klingenberg in: Ann Math 2(69):654–666, 1959). for N-geodesic loops in the reversible Finsler setting.

Delta‐convex structure of the singular set of distance functions

AbstractFor the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

Transnormal Functions and Focal Varieties on Finsler Manifolds

In this paper, we study transnormal functions and their level sets and focal varieties on complete Finsler manifolds. We prove that the focal varieties of a $$C^{2}$$ transnormal function are smooth submanifolds and each regular level set is a tube over either of the focal varieties.

On the Gromov-Hausdorff theory in Finsler geometry

This paper is devoted to the introduction of the Gromov-Hausdorff theory in Finsler geometry. Firstly, we are able to find a special kind of irreversible metric spaces, named as forward metric spaces, which can accurately describe forward complete Finsler manifolds. Secondly, we establish/generalize the Gromov-Hausdorff theory on such metric spaces. In particular, Gromov's precompactness 定理 remains valid in this setting. Last but not least, we obtain a series of precompactness theorems and stable results in the framework of Finsler geometry. Besides, many interesting examples are presented in this paper, which deeply point out genuine differences between Finsler geometry and Riemannian geometry.

Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

Sufficient Criteria for Obtaining Hardy Inequalities on Finsler Manifolds

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if \(\rho \) is a nonnegative function and \(-\varvec{\Delta } \rho \ge 0\) in weak sense, where \(\varvec{\Delta }\) is the Finsler–Laplace operator defined by \( \varvec{\Delta } \rho = \mathrm {div}(\varvec{\nabla } \rho )\), then we obtain the generalization of some Riemannian Hardy inequalities given in D’Ambrosio and Dipierro (Ann Inst H Poincaré Anal Non Linéaire 31(3):449–475, 2014). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo–Nirenberg inequality and a Heisenberg–Pauli–Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler–Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy inequality to a class of Finsler manifolds of bounded geometry.

Affine functions and Busemann functions on complete Finsler manifolds

We clarify the relation between an affine function and a Busemann function in a geodesically complete Finsler manifold. As an application, we give the characterization of a Minkowski space by means of the dimension of the vector space consisting of all affine functions on a Finsler manifold.

Sharp uncertainty principles on general Finsler manifolds

The paper is devoted to sharp uncertainty principles (Heisenberg-Pauli-Weyl, Caffarelli-Kohn-Nirenberg, and Hardy inequalities) on forward complete Finsler manifolds endowed with an arbitrary measure. Under mild assumptions, the existence of extremals corresponding to the sharp constants in the Heisenberg-Pauli-Weyl and Caffarelli-Kohn-Nirenberg inequalities fullycharacterizesthe nature of the Finsler manifold in terms of three non- Riemannian quantities, namely, itsreversibilityand the vanishing of theflag curvatureandSS-curvatureinduced by the measure, respectively. It turns out in particular that the Busemann-Hausdorff measure is the optimal one in the study of sharp uncertainty principles on Finsler manifolds. The optimality of our results are supported by Randers-type Finslerian examples originating from the Zermelo navigation problem.

On the growth rate of geodesic chords

We show that every forward complete Finsler manifold of infinite fundamental group and not homotopy-equivalent to S1 has infinitely many geometrically distinct geodesics joining any given pair of points p and q. In the special case in which β1(M;Z)≥1 and M is closed, the number of geometrically distinct geodesics between two points grows at least logarithmically.

THE SCHWARZIAN DERIVATIVE AND CONFORMAL TRANSFORMATION ON FINSLER MANIFOLDS

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.

A Toponogov-Type Comparison Theorem for Finsler Manifolds and its Applications

In this paper, we introduce the notion of modified flag curvature for Finsler metrics which naturally come from Hessian comparison theorem. We then establish a Toponogov-type comparison theorem for Finsler manifolds with nonnegative-modified flag curvature. As its applications, we prove that the fundamental group of any forward complete Finsler manifold (M, F) with nonnegative-modified flag curvature is finitely generated; furthermore, if additional M has bounded Cartan tensor, then M is of finite topological type.

Some Liouville Theorems on Finsler Manifolds

We give some Liouville type theorems of L p harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014).

Affine vector fields on Finsler manifolds

We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total Ricci curvature.

Strongly and weakly affine vector fields on Finsler manifolds

In this paper, we investigate the affine vector fields on both compact and forward complete Finsler manifolds. We first give definitions of the affine transformation and the affine vector field. Unexpectedly, we find two kinds of affine fields, which are named as the strongly and weakly affine vector fields. Based on these definitions, we prove some rigidity theorems of affine fields on compact and forward complete Finsler manifolds.

Interpolation between Brezis–Vázquez and Poincaré inequalities on nonnegatively curved spaces: sharpness and rigidities

This paper is devoted to investigate an interpolation inequality between the Brezis–Vázquez and Poincaré inequalities (shortly, BPV inequality) on nonnegatively curved spaces. As a model case, we first prove that the BPV inequality holds on any Minkowski space, by fully characterizing the existence and shape of its extremals. We then prove that if a complete Finsler manifold with nonnegative Ricci curvature supports the BPV inequality, then its flag curvature is identically zero. In particular, we deduce that a Berwald space of nonnegative Ricci curvature supports the BPV inequality if and only if it is isometric to a Minkowski space. Our arguments explore fine properties of Bessel functions, comparison principles, and anisotropic symmetrization on Minkowski spaces. As an application, we characterize the existence of nonzero solutions for a quasilinear PDE involving the Finsler–Laplace operator and a Hardy-type singularity on Minkowski spaces where the sharp BPV inequality plays a crucial role. The results are also new in the Riemannian/Euclidean setting.

Remarks on Liouville-type theorems on complete noncompact Finsler manifolds

We give a gradient estimate of the positive solution to the equation ∆u = −λ 2u, λ ≥ 0 on a complete noncompact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete noncompact Finsler manifold we also prove a Liouville-type theorem for a C2 -nonnegative function f satisfying ∆f ≥ cfd, c > 0, d > 1, which improves a result obtained by Yin and He.

A generalization and short proof of a theorem of Hano on affine vector fields

We prove that a bounded affine vector field on a complete Finsler manifold is a Killing vector field. This generalizes the analogous result of Hano for Riemannian manifolds [3]. Even though our result is more general, the proof is significantly simpler.

Frankel's refinement of Bochner's theorem on forward complete Finsler manifolds

In this paper, we prove the Frankel's refinement of Bochner's theorem on both compact and forward complete Finsler manifolds. That is, on a compact or forward complete Finsler manifold M0 with non-positive flag curvature and negative (mean) Ricci curvature, any isometry ψ:M0→M0 homotopic to the identity must be the identity itself.

Strictly convex functions on complete Finsler manifolds

The purpose of the present paper is to investigate the influence of strictly convex functions on the metric structures of complete Finsler manifolds. More precisely we discuss the properties of the group of isometries and the exponential maps on a complete Finsler manifold admitting strictly convex functions.

On a class of complete Finsler manifolds

Here, it is shown that if a forward geodesically complete Finsler manifold admits a circle preserving change of metric then its indicatrix is conformally diffeomorphic to the Euclidean sphere Sn-1. Moreover, if the Finsler manifold is absolutely homogeneous and of scalar flag curvature then it is a Riemannian manifold of constant sectional curvature. These results provide a geometric interpretation for existence of solutions to the certain ODE on the Riemannian tangent space.