Finsleroid-Finsler Space of Involutive Case

TL;DR

This paper constructs and analyzes a specific class of Finsler spaces called Finsleroid-Finsler spaces under an involutive condition, providing explicit formulas for tensors and spray coefficients, and exploring their geometric properties.

Contribution

It introduces the involutive case of Finsleroid-Finsler spaces, deriving explicit tensor and spray coefficient formulas, and clarifying the geometric structure under this condition.

Findings

Explicit formulas for tensors and spray coefficients in the involutive case.

The involutive condition leads to a vanishing normalized tensor derivative.

Derived explicit expressions for the derivative and curvature tensors assuming a parallel 1-form.

Abstract

The Finsleroid-Finsler space is constructed over an underlying Riemannian space by the help of a scalar $g(x)$ and an input 1-form $b$ of unit length. Explicit form of the entailed tensors, as well as the respective spray coefficients, is evaluated. The involutive case means the framework in which the characteristic scalar $g(x)$ may vary in the direction assigned by $b$, such that $dg=\mu b$ with a scalar $\mu(x)$. We show by required calculation that the involutive case realizes through the $A$-special relation the picture that instead of the Landsberg condition $\dot A_{ijk}=0$ we have the vanishing $\dot{\al}_{ijk}=0$ with the normalized tensor $\al_{ijk}=A_{ijk}/||A||$. Under the involutive condition, the derivative tensor $A_{i|j}$ and the curvature tensor $R^i{}_k$ have explicitly been found, assuming the input 1-form $b$ be parallel. Key words: Finsler metrics, spray coefficients, curvature tensors.