Narrow Results

We prove that projective equivalence of two Randers Finsler metrics and such that at least one of the one-forms ω and is not closed implies that for a certain constant C > 0 we have and the form is closed. As an application we prove the natural generalization of the projective Lichnerowicz–Obata conjecture for Randers metrics.

The warped product structures of Finsler metrics are studied in this paper. We give the formulae of the flag curvature and Ricci curvature of these metrics, and obtain the characterization of such metrics to be Einstein. Some Einstein Finsler metrics of this type are constructed.

In this paper, it is shown that the reversible Blaschke Finsler metrics on spheres with vanishing $S$-curvature are Riemannian.

In this paper, we show that a Finsler warped product metric is of almost vanishing $H$-curvature if and only if it is of almost vanishing $\chi$-curvature. Furthermore, the corresponding one form is exact.

A Finsler metric is geodesically reversible if geodesic remain geodesics after a change of orientation. For a reversible Minkowski space, we alter the geodesically reversible Finsler metric without conjugate points in a strictly convex compact subset and show that its central symmetrization is geodesically conjugate to the original reversible Minkowski norm.

We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted — broken — as they pass through the interface, due to the discontinuity of their velocities. This “breaking” must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell’s law. In the same way, optimal paths bouncing off the interface — without crossing into the second domain — provide the generalized law of reflection. The classical Snell’s and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci — stemming from the self-intersection of the wavefronts — which typically appear when they approach a high-speed profile domain from a low-speed profile.

We prove optimal systolic inequalities on Finsler Möbius bands relating the systole and the height of the Möbius band to its Holmes–Thompson volume. We also establish an optimal systolic inequality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Möbius band and the Klein bottle are also presented.

A Finsler space (M, Σ) is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true.

In this note, building on recent work of LeBrun and Mason [18], it is shown that a geodesically reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily projectively flat.

As a corollary, using a previous result of the author [5], it is shown that a reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily a Riemannian metric of constant Gauss curvature, thus settling a long-standing problem in Finsler geometry.

We define the flag curvature of the dual of a Finsler metric and study its relationship with the flag curvature of the initial Finsler metric via Legendre transformation.