Flat surfaces in the Euclidean space \(\mathbb E^{3}\) with pointwise 1-type Gauss map (Q708461)

A submanifold of a Euclidean space \(E^n\) is said to have pointwise 1-type Gauss map if its Gauss map \(G\) satisfies \(\Delta G= f(G+ C)\) for some smooth function \(f\) and a constant vector \(C\). It is said to be of the second kind if \(C\neq 0\). In this paper it is shown that a flat surface in \(E^3\) with pointwise 1-type Gauss map of the second kind is either a right circular cone, a plane or a cylinder with a special base curve the curvature of which satisfies a specific differential equation. In particular there are no tangent surfaces of such a kind.