Communications in Analysis and Geometry

Volume 31 (2023)

Number 9

Rotational surfaces with second fundamental form of constant length

Pages: 2227 – 2254

DOI: https://dx.doi.org/10.4310/CAG.2023.v31.n9.a3

Authors

Alexandre Paiva Barreto (Departamento de Matemática, Universidade Federal de São Carlos, SP, Brazil)

Francisco Fontenele (Departamento de Geometria, Universidade Federal Fluminense, Niterói, RJ, Brazil)

Luiz Hartmann (Departamento de Matemática, Universidade Federal de São Carlos, SP, Brazil)

Abstract

We obtain an infinite family of complete non embedded rotational surfaces in $\mathbb{R}^3$ whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homothety and a rigid motion, a member of that family. In particular, the round sphere and the circular cylinder are the only complete embedded rotational surfaces in $\mathbb{R}^3$ with second fundamental form of constant length.

Received 21 August 2018

Accepted 19 October 2021

Published 12 August 2024