Umbilicity of surfaces with orthogonal asymptotic lines in \(\mathbb R^ 4\). (Q1608973)

The authors study some properties of surfaces \(M\) in 4-space \(\mathbb{R}^4\) all whose points are umbilic with respect to some normal field. In particular, they show that this condition is equivalent to the orthogonality of the globally defined fields of asymptotic directions. The authors also analyze necessary and sufficient conditions for the hypersphericity of surfaces in 4-space \(\mathbb{R}^4\).