CR-structures of codimension 2 on tangent bundles in Riemann-Finsler geometry

TL;DR

This paper characterizes a specific CR-structure of codimension 2 on the tangent bundle of Finsler manifolds, linking it to scalar flag curvature and constant curvature in Riemannian cases.

Contribution

It introduces a new CR-structure on tangent bundles of Finsler manifolds, generalizing previous structures and connecting them to curvature conditions.

Findings

CR-structure exists for scalar flag curvature Finsler manifolds

In Riemannian case, structure relates to constant curvature

Generalization involves a positive parameter with complex conditions

Abstract

We determine a 2-codimensional CR-structure on the slit tangent bundle $T_0M$ of a Finsler manifold $(M, F)$ by imposing a condition regarding the almost complex structure $\Psi$ associated to $F$ when restricted to the structural distribution of a framed $f$-structure. This condition is satisfied when $(M, F)$ is of scalar flag curvature (particularly flat) and in the Riemannian case $(M, g)$ this last condition means that $g$ is of constant curvature. This CR-structure is finally generalized by using one positive number but under more difficult conditions.