A surface which has a family of geodesics of curvature (Q880239)

Let \(S\) be a surface in \(\mathbb{R}^{3}\) and \(\text{Umb} (S)\) the set of umbilical points of \(S\). Let \(\mathcal{D}\) be a smooth one-dimensional \(\mathcal{D}\) distribution on \(S\backslash\text{Umb}(S)\) whose tangent vector has principal direction. Such distribution \(\mathcal{D}\) is called principal distribution on \(S\) and each integral curve of principal distribution is called a line of curvature of \(S\). The first purpose of this paper is an intrinsic characterization of a line of curvature of \(S\). The author characterizes the semisurface structure of \(S\) (S equipped with \(\mathcal{D}_{1}\), \(\mathcal{D}_{2}\) orthogonal one dimensional distributions to each other at any point of \(S\)) in terms of local representation of the first fundamental form (theorem 1 and theorem 2). The second purpose is an extrinsic characterization. He characterizes the curvature and the torsions of the lines of curvature of \(S\) as space curves (theorem 3 and theorem 4).