A Schur Type Lemma for the Mean Berwald Curvature in Finsler Geometry

On two curvature-driven problems in Riemann–Finsler geometry

This article uses the Berwald connection exclusively, together with its two curvatures, to cut an efficient path across the landscape of Finsler geometry. Its goal is to initiate differential geometers into two key research areas in the field: the search for unblemished “unicorns” and the study of Ricci flow. The exposition is almost self-contained.

Toward a Gravitation Theory in Berwald–Finsler Space

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric i...

On the k-nullity foliations in Finsler geometry

Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Unicorns in Finsler Geometry

By unicorns, I am referring to those mythical single-horned horse-like creatures for which there are only rumoured sightings by a privileged few. A similar situation exists in Finsler differential geometry. There, one has the hierarchy Euclidean ⊂ Minkowskian & Riemannian ⊂ Berwald ⊂ Landsberg among five families of metrics, in which the first two inclusions are known to be proper by virtue of ...