Hypersurfaces with parallel cubic form in affine space
Let \(M_{n-1}\) be a hypersurface in an affine \(n\)-dimensional space \(A_ n\), and let \(\nabla\), \(\nabla^ \perp\) and \(\overline{\nabla} = \nabla \oplus \nabla^ \perp\) be connections in the tangent bundle, the normal bundle and the van der Waerden-Bortolotti connection, respectively. If \(\alpha_ 2\) is the second fundamental form of the hypersurface \(M_{n-1}\), then its third fundamental form \(\alpha_ 3 = \overline{\nabla} \alpha_ 2\). If \(\overline {\nabla} \alpha_ 3 = 0\), then the hypersurface \(M_{n-1}\) is said to have parallel third fundamental form. In the paper under review, the author finds a local structure of a tangentially nondegenerate hypersurface \(M_{n-1}\) with parallel third fundamental form with respect to the Blaschke normals. He proves that locally such a hypersurface coincides with a certain improper affine hypersphere.
scientific article; zbMATH DE number 746049