On Doubly Warped Product Finsler Manifolds

TL;DR

This paper introduces and analyzes the geometric properties of doubly warped product Finsler manifolds, establishing conditions for Riemannian structure, curvature relations, and complex structures, advancing the understanding of their geometric and topological features.

Contribution

It defines doubly warped product Finsler manifolds, proves their Riemannian nature under certain conditions, and explores curvature, metric, and complex structures, providing new insights into their geometry.

Findings

Doubly warped product Finsler manifolds are Riemannian if they are C-reducible or proper Berwaldian.

Relations between Riemannian curvatures of the product and its components are established.

Conditions for flatness, scalar flag curvature, and totally geodesic tangent bundles are derived.

Abstract

In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. Also, we obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the slit tangent bundle of a doubly warped product Finsler manifold.