Non-degenerate hypersurface immersions in spheres (Q2781476)

In this doctoral thesis hypersurfaces of the Euclidean unit sphere \(\mathbb{S}^{n+1}\) are treated which are non-degenerate (i.e., their shape operator is of maximal rank). Their first, second and third fundamental forms \(I\), \(II\) and \(III\) are (semi)-Riemannian metrics with corresponding Levi-Civita connections \(\nabla^I\), \(\nabla^{II}\) resp. \(\nabla^{III}\), and \((\nabla^I,II, \nabla^{III})\) is a conjugate tripel in the sense of affine geometry. Motivated by this fact the author considers the difference tensor \(2C =\nabla^I -\nabla^{III}\) and its ``traceless part'' \(\widetilde C\). While \(C \equiv 0\) is equivalent to the parallelity of the shape operator, the conditions \(\widetilde C\equiv 0\) and \(\text{div} \widetilde C\equiv 0\) define two new classes of hypersurfaces, for which a detailed investigation is presented. Non-degenerate intersections of \(\mathbb{S}^{n+1}\) and hyperquadrics \(\subset \mathbb{R}^{n+2}\) with midpoint 0 are proved to be \((\widetilde C\equiv 0)\)-hypersurfaces. The \((\widetilde C\equiv 0)\)-surfaces of \(\mathbb{S}^3\) are classified. Furthermore, the author obtains results on \((\text{div} \widetilde C\equiv 0)\)-hypersurfaces, the so-called Chebyshev hypersurfaces, which are locally symmetric resp. have constant Gauss curvature or constant mean curvature resp. whose shape operator has constant length.