Rotational Surfaces with second fundamental form of constant length

Alexandre P. Barreto, Francisco Fontenele, Luiz Hartmann

Published 2018-12-20Version 1

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.