Natural lifts and curvatures, arc-lengths of the spherical indicatries of the evolute curve in E^3

Abstract

In this paper, we investigated the natural lifts and curvatures of the spherical indicatries of the evolute in E 3 . Firstly, it is shown that the Darboux vektor of evolute curve and binormal vektor of the evolute curve are linearly dependent, secondly the relations among the geodesic curvatures and arc-lengths with respect to E 3 and S 2 of the fixed pole curve (C ∗ ) and the spherical indicartix curves (T ∗ ),(N ∗ ),(B ∗ ) generated by the unit vektor (C ∗ )o nS 2 have been obtained and finally, the condition being the natural lifts of the spherical indicatrix of the evolute curve (α ∗ ) are an integral curve of geodesic spray has expressed.