The book “Handbook of Finsler geometry” has been included with a CD containing an elegant Maple package, FINSLER, for calculations in Finsler geometry. Using this package, an example concerning a Finsler generalization of Einstein’s vacuum field equations was treated. In this example, the calculation of the components of the hv-curvature of Cartan connection leads to wrong expressions. On the o...

Abstract 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 extend results about generic Riemannian metrics to Finsler metrics. show a bumpy theorem for and prove that $$C^4$$ C</mml:mi...

In this paper‎, ‎the‎ ‎authors prove that a strictly Kähler-Berwald manifold with‎ ‎nonzero constant holomorphic sectional curvature must be a‎ Kähler manifold‎. 

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one. For the critical dimensions, we prove also the existenc...

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S → lR (n ≥ 2, S is the circle). We study geometries on the manifold of curves, provided by Sobolev–type Riemannian metrics H . These metrics have been shown to regularize gradient flows used in Computer Vision applications, see [13, 16] and references therein. We provide some basic results of H metrics; and, fo...

A generalization of Finsler structures on surfaces is proposed and the differential invariants of such structures are developed. The information obtained is then used to construct examples of Finsler structures and generalized Finsler structures which satisfy various interesting curvature conditions. In particular, examples are constructed of non-Riemannian Finsler structures on the 2-sphere wh...

We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, and other properties follow immediately from the known results about the cut ...

This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be smooth everywhere is replaced by a s...

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the non-negative weighted Ricci curvature and the convexity of another weight function used in the definition o...

Methods of estimating (Riemannian and Finsler) filling volumes by using nonexpanding maps to Banach spaces of L∞-type are developed and generalized. For every Finsler volume functional (such as the Busemann volume or the Holmes– Thompson volume), a natural extension is constructed from the class of Finsler metrics to all Lipschitz metrics, and the notion of area is defined for Lipschitz surface...