We first give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space R and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, M is diffeomorphic and almost isometric to an n-dimensional sphere.

Landsberg’s notion of disorder, entropy normalized to maximum entropy, was originally proposed for the Shannon information-theoretic entropy to overcome deficiencies of entropy as a measure of disorder due to extensivity. We generalize Landsberg’s concept to three classes of extended entropies: Rényi, Tsallis and Landsberg-Vedral. Three examples are treated, including one based on the logistic ...

The complexity measure from Shiner et al. [Physical Review E 59, 1999, 1459–1464] (henceforth abbreviated as SDL-measure) has recently been the subject of a fierce debate. We discuss the properties and shortcomings of this measure, from the point of view of our recently constructed fundamental, statistical mechanics-based measures of complexity Cs (γ, β) [Stoop et al., J. Stat. Phys. 114, 2004,...

In this paper we take a close look at Lie derivatives on a Finsler bundle and give a geometric meaning to the vanishing of the mixed curvature of certain covariant derivatives on a Finsler bundle. As an application, we obtain some characterizations of Landsberg manifolds.

In this paper we study some rigidity properties for Finsler manifolds of sectional flag curvature. We prove that any Landsberg manifold of non-zero sectional flag curvature and any closed Finsler manifold of negative sectional flag curvature must be Riemannian.

In the present paper, we shall prove new characterizations of Berwald spaces and Landsberg spaces. The main idea inthis research is the use of the so-called average Riemannian metric.