Minimal and pseudo-umbilical rotational surfaces in Euclidean space \(\mathbb E^4\) (Q1943596)

Assume that \(M\) is a general rotational surface in \(\mathbb E^4\) with planar meridian curves. Such a surface has the parametrization: \[ F(s,t)= (x(s)\cos(at), x(s)\sin(at), z(s)\cos(bt), z(s)\sin(bt)), \] where \(a, b\) are constant values that represent the rates of rotation. The paper deals with a characterization of surfaces \(M\) as above in the following cases: {\parindent=6mm \begin{itemize}\item[-] \(M\) is a minimal non planar rotating surface in the Euclidean space \(\mathbb E^4\). \item[-] \(M\) is pseudo-umbilical. \end{itemize}}In both cases, the authors find the general solutions \(x, z\) of the differential equations that translate the given geometric conditions. The main novelty is that different rotation rates \(a, b\) are allowed.