Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces (Q1890834)

In this paper, we investigate a special class of homogeneous locally strongly convex affine hypersurfaces \(M^ n\) in \(\mathbb{R}^{n+1}\): namely those hypersurfaces for which the affine shape operator \(S\) has an eigenvalue \(\lambda\) with multiplicity \((n - 1)\). Such hypersurfaces are called quasi-umbilical. In the case that \(\lambda = 0\), a complete classification was obtained in [\textit{F. Dillen} and \textit{L. Vrancken}, Math. Z. 212, No. 1, 61-72 (1993; Zbl 0804.53014)]. In the present paper, we first prove that a locally strongly convex, locally homogeneous quasi-umbilical hypersurface in \(\mathbb{R}^{n+1}\) satisfies \(\text{det }S= 0\). We then use the above quasi-umbilical hypersurfaces in \(\mathbb{R}^ 4\) and \(\mathbb{R}^ 5\). We also give some more examples of such hypersurfaces in \(\mathbb{R}^{n+1}\).