Ruled surfaces and tangent bundle of unit 2-sphere

Abstract

In this paper, a one-to-one correspondence is given between the tangent bundle of unit 2-sphere, [Formula: see text], and the unit dual sphere, [Formula: see text]. According to Study’s map, to each curve on [Formula: see text] corresponds a ruled surface in Euclidean 3-space, [Formula: see text]. Through this correspondence, we have corresponded to each curve on [Formula: see text] a unique ruled surface in [Formula: see text]. Moreover, the relationships between the developability conditions of these ruled surfaces and their striction curves are analyzed. It is shown that the ruled surfaces corresponding to the involute–evolute curve couples on [Formula: see text] are developable.