Finsleroids with three axes in dimension $N=3$

TL;DR

This paper establishes a comprehensive theorem describing how three-axes Finsleroid metric functions depend on the azimuthal angle in three dimensions, providing explicit solutions under certain conditions.

Contribution

It presents a complete set of algebraic and differential equations characterizing three-axes Finsleroid metrics and derives explicit solutions for their dependence on the azimuthal angle.

Findings

Derived necessary and sufficient conditions for Finsleroid metric functions

Obtained explicit dependence of metric functions on the azimuthal angle

Established a complete mathematical framework for three-axes Finsleroids

Abstract

The Conclusive Theorem has been established to determine the dependence of the three-axes positive-definite Finsleroid metric functions $F$ on the Finsleroid azimuthal angle $\theta$ in the three-dimensional case $N=3$, provided that the condition of the angle-separation in the involved characteristic functions is implied. The complete set of algebraic and differential equations is derived in all rigor which are necessary and sufficient in order that the function $F$ belong to the class. It proves possible to solve the equations and obtain the explicit dependence of the involved characteristic functions on the angle $\theta$.