# Rotational Surfaces with second fundamental form of constant length

## TL;DR

This paper classifies complete rotational surfaces in three-dimensional space with a second fundamental form of constant length, identifying a family of such surfaces and characterizing the only embedded examples as spheres and cylinders.

## Contribution

It introduces an infinite family of non-embedded rotational surfaces with constant-length second fundamental form and characterizes all complete rotational surfaces with this property.

## Key findings

- Complete non-embedded rotational surfaces with constant second fundamental form form an infinite family.
- Complete embedded rotational surfaces with this property are only spheres and cylinders.
- The classification includes a unique family up to homothety and rigid motions.

## 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.