We characterize compact metric spaces whose locally flat Lipschitz functions separate points uniformly as exactly those that are purely 1-unrectifiable, resolving a problem of Weaver. subsequently use this geometric characterization to answer several questions in analysis. Notably, it follows the Lipschitz-free space $\mathcal{F}(M)$ over $M$ is dual if and only 1-unrectifiable. Furthermore, we...

In this paper we study the properties of special (α, β)-metric α α−β + β, the Randers change of Matsumoto metric. We find a necessary and sufficient condition for this metric to be of locally projectively flat and we prove the conditions for this metric to be of Berwald and Douglas type.

Dually flat manifolds constitute fundamental mathematical objects of information geometry. This note establishes some facts on the global properties and topology of dually flat manifolds which, in particular, provide answers to questions and problems in global information geometry posed by Amari and Amari-Nagaoka.

We exactly solve a special matrix model of dually weighted planar graphs describing pure two-dimensional quantum gravity with a R interaction in order to study the intermediate regimes between the gravitating and flat metric. The flat space is modeled by a regular square lattice, while localized curvature is being introduced through defects of the lattice. No “flattening” phase transition is fo...

We develop a family of infinite-dimensional (i.e. non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class Ck b with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and the dually flat geometry of Amari...

We study Ricci flat 4-metrics of any signature under the assumption that they allow a Lie algebra of Killing fields with 2-dimensional orbits along which the metric degenerates and whose orthogonal distribution is not integrable. It turns out that locally there is a unique (up to a sign) metric which satisfies the conditions. This metric is of signature (++−−) and, moreover, homogeneous possess...

the main objective of this paper is to find the necessary and sufficient condition of a given finslermetric to be einstein in order to classify the einstein finsler metrics on a compact manifold. the consideredeinstein finsler metric in the study describes all different kinds of einstein metrics which are pointwiseprojective to the given one. this study has resulted in the following theorem tha...

In this paper, we study the Cartan space with some (α, β) metrics, in particular Randers metric admitting h-metrical d-connection. Further, we show that the condition for Cartan space with Randers metric to be locally Minkowski and Conformally flat. 2000 Mathematics Subject Classification: 53B40, 53C60

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n ≥ 3. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.