On general helices and pseudo-riemannian manifolds

Year 1998, Volume: 47 , 0 - 0, 01.01.1998

N. Ekmekçi

https://doi.org/10.1501/Commua1_0000000404

Abstract

In a Riemannian manifold, a regular curve is called a general helix if is constant and its firs and second curvatures are not constant [4]. İf its First and second curvatures are constant the third curvature is zero then the regular curve is called helix. For helices in a Lorentzian manifold, there is a research of T. Ikawa, who investigated and obtained the differential equation;
D D D X = KD X , (K = a - p5
XXX X
fOT the drcular helix which corresponds to the case that the curvatures a and P of the timelike curve c(t) on the Lorentzian manifold M are constant [3], Later, N. Ekmekçi and H.H. HacısaUhoğlu obtained the differential equation I\I\DxX = KD^K + 3a' D^Y ,
K = of + a2 P')
P
fcff the case of general helix [2]. Recently, T. Nakanishi [5] prove the following lemma about a helix in Pseudo-Riemannian manifold which is stated as, “A unit speed curve c in M is a helix if and only if there exist a constant X such that D D D X = XD X”
XXX X
a
îhis paper generalizes the lemma stated above lo the case of a general helix.

Keywords

general helices, pseudo-riemannian, manifolds

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi