ON CHARACTERIZATIONS OF SPHERICAL CURVES

Haila Alodan (Saudi Arabia) and Hana Al-Sodais (Saudi Arabia)

Received September 17, 2007

Abstract

A curve is said to be a spherical curve if it lies on a sphere. There are many characterizations of spherical curves, among those is the well-known theorem of Breuer and Gottlieb [4], which was then improved by Wong without any preconditions. It states that a -curve lies on a sphere if and only if where A, B are arbitrary constants and k is the curvature of the curve and t is the torsion of the curve. This result was then generalized by Alodan and Deshmukh to n-dimensional submanifolds of a Euclidean space  which states that an n-dimensional compact connected and oriented submanifold of  lies on a hypersphere if and only if  where H is the mean curvature vector field and  is the normal component of the position vector field y of the submanifold in  In this paper we prove the equivalence of Wong’s result with Alodan and Deshmukh’s result in two ways.

Keywords and phrases: spherical curves, submanifolds, mean curvature.