Finsleroid--Finsler Space with Berwald and Landsberg Conditions

TL;DR

This paper introduces the Finsleroid--Finsler space, exploring its geometric properties, conditions for being Berwald or Landsberg types, and the associated curvature tensor structures.

Contribution

It formulates the Finsleroid--Finsler space concept, derives explicit conditions for Landsberg and Berwald types, and analyzes the curvature tensor structures.

Findings

Finsleroid charge is constant in Landsberg type spaces.

Finsleroid--Finsler space is Berwald if the Finsleroid--axis 1-form is parallel and charge is constant.

Explicit conditions for Landsberg type are derived.

Abstract

We formulate the notion of the Finsleroid--Finsler space, including the positive--definite as well as indefinite cases. The associated concepts of angle, scalar product, and the distance function are elucidated. If the Finsleroid--Finsler space is of Landsberg type, then the Finsleroid charge is a constant. The Finsleroid--Finsler space proves to be a Berwald space if and only if the Finsleroid--axis 1-form is parallel with respect to the associated Riemannian metric and, simultaneously, the Finsleroid charge is a constant. The necessary and sufficient conditions for the Finsleroid--Finsler space to be of the Landsberg type are found, which are explicit and simple. The structure of the associated curvature tensors has been elucidated.