Öz
Kaynakça
- [1] A. J. C. Barré de Saint-Venant, Mémoire sur les lignes courbes non planes, Journ. Ec. Polyt. 30, 1–76, 1846.
- [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] B. Y. Chen, Differential geometry of rectifying submanifolds, Int. Electron. J. Geom. 9, 1–8, 2016.
- [4] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math. 48, 209–214, 2017.
- [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] A. J. Di Scala and G. Ruiz-Hernández, Helix submanifolds of Euclidean spaces, Monasth Math. 157, 205–215, 2009.
- [7] A. Fialkow, Conformals geodesics, Trans. Amer. Math. Soc. 45 (3), 443–473, 1939.
- [8] S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28, 153–163, 2004.
- [9] I. B. Kim, Special concircular vector fields in Riemannian manifolds, Hirosima Math. J. 12, 77–91, 1982.
- [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] M. I. Munteanu, From golden spirals to constant slope surfaces, J. Math. Phys. 51, 073507, 2010.
- [12] P. D. Scofield, Curves of constant precession, Amer. Math. Monthly 102, 531–537, 1995.
- [13] D. J. Struik, Lectures on Classical Differential Geometry, Dover, New York, 1988.
- [14] K. Yano, Concircular geometry I, concircular transformations, Proc. Imp. Acad. Tokyo 16, 195–200, 1940.
Öz
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.
Anahtar Kelimeler
generalized helix slant helix rectifying curve concircular helix generalized cylinder helix surface conical surface concircular surface tangent surface
Kaynakça
- [1] A. J. C. Barré de Saint-Venant, Mémoire sur les lignes courbes non planes, Journ. Ec. Polyt. 30, 1–76, 1846.
- [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] B. Y. Chen, Differential geometry of rectifying submanifolds, Int. Electron. J. Geom. 9, 1–8, 2016.
- [4] B. Y. Chen, Rectifying curves and geodesics on a cone in the Euclidean 3-space, Tamkang J. Math. 48, 209–214, 2017.
- [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] A. J. Di Scala and G. Ruiz-Hernández, Helix submanifolds of Euclidean spaces, Monasth Math. 157, 205–215, 2009.
- [7] A. Fialkow, Conformals geodesics, Trans. Amer. Math. Soc. 45 (3), 443–473, 1939.
- [8] S. Izumiya and N. Takeuchi, New special curves and developable surfaces, Turk. J. Math. 28, 153–163, 2004.
- [9] I. B. Kim, Special concircular vector fields in Riemannian manifolds, Hirosima Math. J. 12, 77–91, 1982.
- [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] M. I. Munteanu, From golden spirals to constant slope surfaces, J. Math. Phys. 51, 073507, 2010.
- [12] P. D. Scofield, Curves of constant precession, Amer. Math. Monthly 102, 531–537, 1995.
- [13] D. J. Struik, Lectures on Classical Differential Geometry, Dover, New York, 1988.
- [14] K. Yano, Concircular geometry I, concircular transformations, Proc. Imp. Acad. Tokyo 16, 195–200, 1940.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Yayımlanma Tarihi | 15 Ağustos 2023 |
| Yayımlandığı Sayı | Yıl 2023 |