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

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 TpM at an arbitrary point p on M.The physical forces are represented by vector fields. We investigate physical forces whose vectors are in TpM on (M,g,Q). Moreover, these vectors are isotropic and they act along isotropic curves. We study the physical work done by such forces.