Submanifolds in SE(3) and their applications in envelope theory

Abstract

Abstract This paper is mainly divided into three parts. In the first part, we give the descriptions of the bases of two and three dimensional Lie subalgebras of se(3). In the second part, we construct submanifolds of SE(3) via several two and three dimensional Lie subalgebras of se(3). In the final part, we focus on the multi-parameter families of curves in E3 generated by actions of the constructed submanifolds of SE(3) on a segment of curve. The geometrical properties of the envelopes of these families of curves are further studied and the expressions of Gauss and mean curvatures in each case are given.