Ruled surfaces and tubes with finite type Gauss map (Q1321744)

An \(m\)-dimensional, connected submanifold \(M^ m\) of a Euclidean \(n\)- space \(E^ n\) is said to be of finite type if the position vector field \(X\) of \(M^ m\) in \(E^ n\) can be expressed as a finite sum of (vectorial) eigenfunctions of the Laplacian of \(\Delta\) of \(M^ m\), acting on smooth functions on \(M^ m\). Similarly, the Gauss map \(G\) of \(M^ m\) is said to be of finite type if \(G\) is a finite sum of eigenfunctions of \(\Delta\). In the paper under review the authors prove the following theorems: 1. Cylinders over curves of finite type and planes are the only ruled surfaces in \(E^ n\) \((n \geq 3)\) with finite type Gauss map. 2. Circular cylinders are the only tubes in \(E^ 3\) with finite type Gauss map.