Natural lifts and the geodesic sprays for the spherical indicatrıces of the mannheim partner curves in E3

Abstract

In this paper, we investigated the natural lifts and curvatures of the spherical indicatrices of the Mannheim partner curves in . Firstly, it is shown that the Darboux vector of Mannheim curve and tangent vector of the Mannheim partners should be linearly dependent. Secondly, the relations among the geodesic curvatures and arc-lengths with respect to and of the fixed pole curve  and the spherical indicatrix curves generated by the unit vector occured depending on the Frenet frame of Mannheim partner curveon  have been obtained and finally, the condition being that the natural lifts of the spherical indicatrix curves of the Mannheim partner curve are an integral curve of geodesic spray has been expressed depending on Mannheim curve.   Key words: Mannheim curve, natural lift of spherical indicatrix curve, geodesic spray, geodesic curvatures.