An Ap-Structure with Finslerian Flavor: I

TL;DR

This paper introduces a new geometric structure combining absolute parallelism and Finsler properties, with naturally arising connections and conditions linking Finslerian metrics to the structure.

Contribution

It constructs a novel FAP-structure with position and direction dependence, defining associated connections and metric conditions, expanding the geometric framework.

Findings

Defined a FAP-structure with position and direction dependence

Established non-linear, Berwald, and Cartan connections with torsion

Derived conditions for Finslerian metrics and AP-structure

Abstract

A geometric structure (FAP-structure), having both absolute parallelism and Finsler properties, is constructed. The building blocks of this structures are assumed to be functions of position and direction. A non-linear connection emerges naturally and is defined in terms of the building blocks of the structure. Two linear connections, one of Berwald type and the other of the Cartan type, are defined using the non-linear connection of the FAP. Both linear connections are non-symmetric and consequently admit torsion. A metric tensor is defined in terms of the building blocks of the structure. The condition for this metric to be a Finslerian one is obtained. Also, the condition for an FAP-space to be an AP-one is given.