Related papers: Deformations and Hilbert's Fourth Problem

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat…

Locally projectively flat metrics (or sprays) form a rich class of metrics (or sprays) in Finsler and spray geometry. The characterization of such metrics is the Hilbert Fourth Problem in the regular case. In this paper we study the…

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy…

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat…

New mathematical objects called Finslerian N-spinors are discussed. The Finslerian N-spinor algebra is developed. It is found that Finslerian N-spinors are associated with an N^2-dimensional flat Finslerian space. A generalization of the…

This paper explores the generalized projective Riemann curvature in Finsler geometry, focusing on the properties of projectively equivalent Finsler metrics and the invariance of their curvature structures under projective transformations.…

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective…

We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6].

This paper reviews several Riemannian metrics and evolution equations in the context of diffeomorphic shape analysis. After a short review of of various approaches at building Riemannian spaces of shapes, with a special focus on the…

A particular Finsler-metric proposed in [1,2] and describing a geometry with a preferred null direction is characterized here as belonging to a subclass contained in a larger class of Finsler-metrics with one or more preferred directions…

We solve explicitly the geodesic equation for a wide class of (pseudo)-Riemannian homogeneous manifolds (G/H,m), including those with G compact, as well as non-compact semisimple Lie groups, under a simple algebraic condition for the metric…

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial…

In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation…

Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as…

We use the Fr\"olicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use…

In a previous paper, the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kaehler. In this article, we consider general Ricci-flat deformations of such spaces, assuming only…

We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the "link" of the…

We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors,…

In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m\ne -1$) of scalar flag curvature must be locally…

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential…