Concircular helices and concircular surfaces in Euclidean 3-space R^3

Abstract

Given a submanifold $M\subset R^{n}$ and a concircular vector field $Y\in Con(R^{n})$, $M$ is said to be a concircular submanifold (with axis $Y$) if $\langle N,Y\rangle$ is a constant function along $M$, $N$ being any unit vector field in the first normal space. In this paper we characterize concircular helices in $R^3$ by means of a differential equation involving their curvature and torsion. We find a full description of concircular surfaces in $R^3$ as a special family of ruled surfaces, and we show that $M\subset R^{3}$ is a proper concircular surface if and only if either $M$ is parallel to a conical surface or $M$ is the normal surface to a spherical curve. Finally, we characterize the concircular helices as geodesics of concircular surfaces.