The value of the work done by an isotropic vector force field along an isotropic curve

TL;DR

This paper explores the work done by isotropic vector forces along isotropic curves on a 3D manifold with specific geometric structures, linking differential geometry with physical force analysis.

Contribution

It introduces a framework for analyzing physical forces represented by isotropic vectors on manifolds with a Q-structure, combining geometric and physical concepts.

Findings

Characterization of isotropic vectors in the manifold

Analysis of work done by isotropic forces along isotropic curves

Integration of geometric structures with physical force modeling

Abstract

In the present paper we consider a 3-dimensional differentiable manifold $M$ equipped with a Riemannian metric $g$ and an endomorphism $Q$, whose third power is the identity and $Q$ acts as an isometry on $g$. Both structures $g$ and $Q$ determine an associated metric $f$ on $(M, g, Q)$. The metric $f$ is necessary indefinite and it defines isotropic vectors in the tangent space $T_{p}M$ at an arbitrary point $p$ on $M$. The physical forces are represented by vector fields. We investigate physical forces whose vectors are in $T_{p}M$ on $(M, g, Q)$. Moreover, these vectors are isotropic and they act along isotropic curves. We study the physical work done by such forces.