Non-reversible Finsler Research Articles

Two short closed geodesics on a sphere of odd dimension

We show that for an open and dense set of non-reversible Finsler metrics on a sphere S^n of odd dimension n=2m-1ge 3 there is a second closed geodesic with Morse index le 4(m+2)(m-1)+2.

Boundary rigidity for Randers metrics


 If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.

Correction to: Some functional inequalities on non-reversible Finsler manifolds

Correction to: Some functional inequalities on non-reversible Finsler manifolds

Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$

We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on~$S^2$ with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or have constant flag curvature

Minimal Boundaries in Tonelli Lagrangian Systems

Abstract We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, that is, a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $\textrm{T} M$, the union of the supports of all lifted minimal boundaries with a given energy projects injectively to the base $M$. Finally, we prove the existence of action minimizing simple periodic orbits on energies just above the Mañé critical value of the universal abelian cover. This provides in particular a class of nonreversible Finsler metrics on the two-sphere possessing infinitely many closed geodesics.

Some functional inequalities on non-reversible Finsler manifolds

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the $$\Gamma $$ -calculus of Bakry et al (2014), we show the dimensional versions of the Poincare–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti and Mondino (2015) in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

Improved Hardy and Rellich inequalities on nonreversible Finsler manifolds

In this paper, we study the sharp constants of quantitative Hardy and Rellich inequalities on nonreversible Finsler manifolds equipped with arbitrary measures. In particular, these inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement, if Finsler manifolds are of strictly negative flag curvature, vanishing S-curvature and finite uniformity constant. Furthermore, these results remain valid when Finsler metrics are reversible.

On deformations of the spectrum of a Finsler-Laplacian that preserve the length spectrum

In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4\lambda_1 > h^2$, where $\lambda_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow.

Almost isometries of non-reversible metrics with applications to stationary spacetimes

We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endowed with a timelike conformal field K , in which case conformal diffeomorphisms correspond to almost isometries of the Fermat metrics defined in the spatial part. A series of results on the topology and the Lie group structure of conformal maps are discussed.

CLIFFORD-WOLF TRANSLATIONS OF LEFT INVARIANT RANDERS METRICS ON COMPACT LIE GROUPS

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two point $x_1, x_2\in M$ there is a Clifford-Wolf translation $\rho$ such that $\rho(x_1)=x_2$. In this paper, we study Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups. The mian result is that a left invariant Randers metric on a connected compact simple Lie group is Clifford-Wolf homogeneous if and only if the indicatrix of the metric is a round sphere with respect to a bi-invariant Riemannian metric. This presents a large number of examples of non-reversible Finsler metrics which are Clifford-Wolf homogeneous.

On the energy functional on Finsler manifolds and applications to stationary spacetimes

In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows us to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.

Gromov pre-compactness theorems for nonreversible Finsler manifolds

In this paper, we consider general metric spaces and length spaces whose metrics may be nonreversible, and give Gromov pre-compactness theorems for such spaces with finite reversibility with respect to a generalized Gromov–Hausdorff distance. On the basis of these we derive pre-compactness theorems for general Finsler manifolds with finite reversibility.

The second closed geodesic on Finsler spheres of dimension 𝑛>2

We show the existence of at least two geometrically distinct closed geodesics on an n n -dimensional sphere with a bumpy and non-reversible Finsler metric for n > 2. n > 2.

The length of a shortest geodesic loop

We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper to give a geometric characterization of dynamically convex Finsler metrics on the 2-sphere. To cite this article: H.-B. Rademacher, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The second closed geodesic on a complex projective plane

We show the existence of at least two geometrically distinct closed geodesics on a complex projective plane with a bumpy and non-reversible Finsler metric.

Existence of closed geodesics on positively curved Finsler manifolds

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

A sphere theorem for non-reversible Finsler metrics

For a non-reversible Finsler metric F on a compact smooth manifold M we introduce the reversibility λ= max {F(−X)|F(X)=1}≥1. We prove the following generalization of the classical sphere theorem in Riemannian geometry: A simply-connected and compact Finsler manifold of dimension n≥3 with reversibility λ and with flag curvature \({{{{\left({{1-\frac{{1}}{{1+\lambda}}}}\right)}}^2 < K \le 1}}\) is homotopy equivalent to the n-sphere.