On the Geometry of the Indicatrix Bundle of a Finsler Manifold

In this paper, we investigate a class of metrics induced by $F$-natural metrics on the indicatrix bundle of a Finsler manifold. This class constitutes a four-parameter family that generalizes the well-known $g$-natural metrics on the unit tangent bundle of a Riemannian manifold. Within this framework, we construct a three-parameter family of contact metric structures whose associated metrics are $F$-natural, and we establish that, in contrast to the Riemannian case —where all such $K$-contact metrics on unit tangent bundles are necessarily Sasakian— the corresponding structures in the Finslerian setting can be $K$-contact without being Sasakian. Furthermore, we provide a characterization of Finsler manifolds with positive constant flag curvature via the existence of $K$-contact structures on their indicatrix bundles.