Concircular helices and concircular surfaces in Euclidean 3-space $\mathbb{R}^{3}$

Abstract

In this paper we characterize concircular helices in $\mathbb{R}^{3}$ by means of a differential equation involving their curvature and torsion. We find a full description of concircular surfaces in $\mathbb{R}^{3}$ as a special family of ruled surfaces, and we show that $M\subset\mathbb{R}^{3}$ 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.

Keywords

• generalized helix
• slant helix
• rectifying curve
• concircular helix
• generalized cylinder
• helix surface
• conical surface
• concircular surface
• tangent surface

References

1. [1] A. J. C. Barré de Saint-Venant, Mémoire sur les lignes courbes non planes, Journ. Ec. Polyt. 30, 1–76, 1846.
2. [2] B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly 110, 147–152, 2003.
3. [3] B. Y. Chen, Differential geometry of rectifying submanifolds, Int. Electron. J. Geom. 9, 1–8, 2016.
4. [4] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math. 48, 209–214, 2017.
5. [5] M. Crampin, Concircular vector fields and special conformal Killing tensors, in: Differential Geometric Methods in Mechanics and Field Theory, 57–70, Academia Press, Gent, 2007.
6. [6] A. J. Di Scala and G. Ruiz-Hernández, Helix submanifolds of Euclidean spaces, Monasth Math. 157, 205–215, 2009.
7. [7] A. Fialkow, Conformals geodesics, Trans. Amer. Math. Soc. 45 (3), 443–473, 1939.
8. [8] S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28, 153–163, 2004.

9. [9] I. B. Kim, Special concircular vector fields in Riemannian manifolds, Hirosima Math. J. 12, 77–91, 1982.
10. [10] P. Lucas and J. A. Ortega-Yagües, Slant helices in the Euclidean 3-space revisited, Bull. Belg. Math. Soc. Simon Stevin 23, 133–150, 2016.
11. [11] M. I. Munteanu, From golden spirals to constant slope surfaces, J. Math. Phys. 51, 073507, 2010.
12. [12] P. D. Scofield, Curves of constant precession, Amer. Math. Monthly 102, 531–537, 1995.
13. [13] D. J. Struik, Lectures on Classical Differential Geometry, Dover, New York, 1988.
14. [14] K. Yano, Concircular geometry I, concircular transformations, Proc. Imp. Acad. Tokyo 16, 195–200, 1940.