ON THE EXPLICIT CHARACTERIZATION OF CURVES ON A (n - 1)-SPHERE IN Sn

Abstract

In (n+1)-dimensional Euclidean space E-n+(1), harmonic curvatures and focal curvatures of a non-degenerate curve were defined by Ozdamar and Hacisalihoglu in [7] and by Uribe-Vargas in [9], respectively. In this paper, we investigate the relations between the harmonic curvatures of a non-degenerate curve and the focal curvatures of tangent indicatrix of the curve. Also we give the relationship between the Frenet apparatus (vectors and the curvature functions) of a curve alpha in E-n (+1) and the Frenet apparatus of tangent indicatrix alpha(T) of the curve alpha. In the main theorem of the paper, we give a characterization for a curve to be a (n-1)-spherical curve in S-n by using focal curvatures of the curve. Furtermore we give that harmonic curvature of the curve is focal curvature of the tangent indicatrix.