The Chen-type of the spiral surfaces
A spiral surface is generated by rotating a plane curve \(C\) about an axis \(A\) contained in the plane of the curve \(C\) and simultaneously transforming \(C\) homothetically relative to a point of \(A\). In this paper, the authors study the type number and the Gauss map of spiral surfaces and prove the following results: Theorem 1. A spiral surface \(M\) in Euclidean 3-space \(E^3\) is of finite type if and only if \(M\) is minimal. Theorem 2. The plane is the only spiral surface in \(E^3\) with finite type Gauss map. Theorem 3. The plane is the only spiral surface in \(E^3\) whose Gauss map \(G\) satisfies \(\Delta G = \Lambda G\), where \(\Lambda\) is an endomorphism of \(E^3\).
scientific article; zbMATH DE number 828273
scientific article