Torse-forming vector field with certain deformations
Beldjilali Gherici, Bayour Benaoumeur, Bouzir Habib
TL;DR
The paper investigates torse-forming vector fields, a broad generalization of several classical vector fields, and develops a systematic set of metric deformations to convert a proper torse-forming field into special cases. It derives precise conditions under which a torse-forming vector field remains torse-forming, becomes recurrent, or turns concircular under three deformation schemes: conformal, $\mathcal{D}$-isometric, and $\omega$-conformal. Through explicit calculations and examples on spaces like $\mathbb{S}^3$ and $\mathbb{R}^3$, the authors demonstrate how to engineer desired torse-forming behavior via these deformations. The work provides a practical framework for manipulating torse-forming fields in Riemannian geometry with potential implications for relativity and submanifold theory. Overall, it connects geometric function–vector field interplay with structured metric changes to generate targeted special cases.
Abstract
Torse-forming vector fields are generalizations of some important vector fields. In this paper, we present some techniques to transform a proper torse-forming vector field into its special cases. Concrete examples are given.