Cubic form geometry for surfaces in \({\mathbf S}^3(1)\) (Q1610980)

Let \(M^2\) be a nondegenerate surface in a sphere \(S^3(1)\), and \(C\) the difference tensor field between the Levi-Civita connections \(\nabla^I\) and \(\nabla^{III}\) of its first and third fundamental forms, i.e. \(C:= {1\over 2}(\nabla^I- \nabla^{III})\). A local classification is given for nondegenerate surfaces \(M^2\) in \(S^3(1)\subset R^4\) with vanishing traceless part \(\widetilde C\) of \(C\). It is proved that such an \(M^2\) is locally either totally umbilical, or a part of a rotational surface whose profile curve is an arc of an ellipse or of a hyperbola, or a part of a quadratic surface of a special type, which is described in the paper by the explicit parametric equations in \(R^4\).