## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 28, Number 1 (2004), 223-243.

### Parallel curved surfaces

#### Abstract

A surface $S$ in $R^{3}$ 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 [2], we studied real-analytic, parallel curved surfaces and in particular, we showed 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, we shall show 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. However, we shall also show that for each non-negative integer $g\in N\cup\{0\}$, there exists a connected, compact, orientable, embedded, parallel curved surface of genus $g$.

#### Article information

**Source**

Tsukuba J. Math., Volume 28, Number 1 (2004), 223-243.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496164723

**Digital Object Identifier**

doi:10.21099/tkbjm/1496164723

**Mathematical Reviews number (MathSciNet)**

MR2082231

**Zentralblatt MATH identifier**

1073.53003

#### Citation

Ando, Naoya. Parallel curved surfaces. Tsukuba J. Math. 28 (2004), no. 1, 223--243. doi:10.21099/tkbjm/1496164723. https://projecteuclid.org/euclid.tkbjm/1496164723