On Minkowskian Product of Finsler Manifolds

TL;DR

This paper investigates the Minkowskian product of Finsler manifolds, deriving connections and conditions for special properties, providing a method to construct Finsler manifolds with desired geometric features.

Contribution

It introduces a new framework for Minkowskian products of Finsler manifolds and characterizes conditions for them to be Berwald or Landsberg types.

Findings

Derived Cartan and Berwald connections for the product manifold.

Established necessary and sufficient conditions for special Finsler manifold types.

Provided a method for constructing Finsler manifolds with specific properties.

Abstract

Let (M_1,F_1) and (M_2,F_2) be a pair of Finsler manifolds. The Minkowskian product Finsler manifold (M,F) of (M_1,F_1) and (M_2,F_2) with respect to a product function f is the product manifold M=M_1\times M_2 endowed with the Finsler metric F^2=f(K,H), where K=(F_1)^2,H=(F_2)^2. In this paper, the Cartan connection and Berwald connection of (M,F) are derived in terms of the corresponding objects of (M_1,F_1) and (M_2,F_2). Necessary and sufficient conditions for (M,F) to be Berwald (resp. weakly Berwald, Landsberg, weakly Landsberg) manifold are obtained. Thus an effective method for constructing special Finsler manifolds mentioned above is given.