Finsleroid-Space Supplemented by Angle

TL;DR

This paper extends the $\\cE_g^{PD}$-geometry to include a two-vector Finslerian metric function that naturally incorporates angle and scalar product, with explicit solutions derived from geodesic equations, enhancing Finslerian and Minkowskian geometries.

Contribution

It introduces a two-vector extension of the Finslerian metric function in $\\cE_g^{PD}$-space, enabling the definition of angle and scalar product in a natural way.

Findings

Explicit solutions for geodesic equations are obtained.

The angle measure is derived from the geodesic solutions.

The Finsleroid replaces the unit sphere as the carrier of the spherical image.

Abstract

Our previous exploration of the $\cE_g^{PD}$-geometry has shown that the field is promising. Namely, the $\cE_g^{PD}$-approach is amenable to development of novel trends in relativistic and metric differential geometry and can particularly be effective in context of the Finslerian or Minkowskian Geometries. The main point of the present paper is the tenet that the $\cE_g^{PD}$-space-associated one-vector Finslerian metric function admits in quite a natural way an attractive two-vector extension, thereby giving rise to angle and scalar product. The underlying idea is to derive the angular measure from the solutions to the geodesic equation, which prove to be obtainable in an explicit simple form. The respective investigation is presented in Part I. Part II serves as an extended Addendum enclosing the material which is primary for the $\cE_g^{PD}$-space. The Finsleroid, instead of the unit sphere, is taken now as carrier proper of the spherical image. The indicatrix is, of course, our primary tool.