Rotational surfaces with second fundamental form of constant length

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.

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All of the authors are partially supported by CNPq (Brasil).

Received 21 August 2018

Accepted 19 October 2021

Published 12 August 2024