Parallel curved surfaces (Q1769867)

A surface \(S\) in a three-dimensional Euclidean space \(\mathbb{R}^3\) is called parallel curved if there exists a plane such that at each point of \(S\), there exists a principal direction parallel to this plane. In [Tsukuba J. Math. 26, 251--267 (2002; Zbl 1029.53009)], the author studied real-analytic, parallel curved surfaces and, in particular, he proved that a connected, complete, real-analytic, embedded, parallel curved surface is homeomorphic to a sphere, a plane, a cylinder or a torus. In the present paper, he proves that a connected, complete, embedded, parallel curved surface such that any umbilical point is isolated is also homeomorphic to a sphere, a plane, a cylinder or a torus. He also shows that for each non-negative integer \(g \in \mathbb{N} \cup \{0\}\), there exists a connected compact orientable, embedded, parallel curved surface of genus \(g\).