Related papers: Deformations and Hilbert's Fourth Problem

A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first…

We use two non-Riemannian curvature tensors, the $\chi$-curvature and the mean Berwald curvature to characterise a class of Finsler metrics admitting first integrals.

The space of all non degenerate bilinear structures on a manifold $M$ carries a one parameter family of pseudo Riemannian metrics. We determine the geodesic equation, covariant derivative, curvature, and we solve the geodesic equation…

This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…

An extension of Riemmann's geometry into a direction dependent geometric structure is usually described by Finsler's geometry. Historically, this construction was motivated by the well-known Riemann's quartic length element example. Quite…

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…

A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that…

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of…

We define a $\mathbb{Q}$-valued deformation invariant of certain complete Riemann-Finsler manifolds, in particular of complete Riemannian manifolds with non positive sectional curvature. It is proved that every rational number is the value…

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic…

For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics…

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas…

In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops…

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…

The geometry of the $q$-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection…

In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a convex bounded domain of $\mathbb{R}^n$ is the Hilbert geometry in that domain…

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the…

In this paper, we characterize locally dually flat and Antonelli $m$-th root Finsler metrics. Then, we show that every $m$-th root Finsler metric of isotropic mean Berwald curvature reduces to a weakly Berwald metric.

This PhD dissertation covers a range of topics in Finsler geometry and Finsler gravity, most notably: (i) the characterization of Berwald spaces, (ii) pseudo-Riemann (non-)metrizability of Berwald spaces, (iii) $(\alpha,\beta)$-metrics,…

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a manifold of dimension n \geq 3 is either Riemannian or locally…