Doubly Warped Product Finsler Manifolds with Some Non-Riemannian Curvature Properties

TL;DR

This paper investigates the curvature properties of doubly warped product Finsler manifolds, establishing conditions under which they exhibit Riemannian or Landsbergian characteristics and showing the non-existence of locally dually flat proper cases.

Contribution

It provides new insights into the curvature properties of doubly warped product Finsler manifolds, including characterizations of Berwald, Landsberg, and Riemannian cases, and proves the non-existence of locally dually flat proper DWP-Finsler manifolds.

Findings

Proper Douglas DWP-Finsler manifolds are Riemannian.

A DWP-manifold is Landsbergian if and only if it is Berwaldian.

No locally dually flat proper DWP-Finsler manifold exists.

Abstract

In this paper, we consider doubly warped product (DWP) Finsler manifolds with some non-Riemannian curvature properties. First, we study Berwald and isotropic mean Berwald DWP-Finsler manifolds. Then we prove that every proper Douglas DWP-Finsler manifold is Riemannian. We show that a proper DWP-manifold is Landsbergian if and only it is Berwaldian. In continue, we prove that every relatively isotropic Landsberg DWP-manifold is a Landsberg manifold. We show that a relatively isotropic mean Landsberg warped product manifold is a weakly Landsberg manifold. Finally, we show that there is no any locally dually flat proper DWP-Finsler manifold.