Flat hyperbolic Centro-affine Tchebychev hypersurfaces of \(\mathbb{R}^4\)
Let \(M\) be a hyperbolic centro-affine hypersurface in \(\mathbb{R}^4\), and let \(X, Y, Z\) be vector fields in \(T_pM\). Assume that \(M\) is flat, in other words, that the curvature tensor \(R\) associated with the Levi-Civita connection of the centro-affine metric satisfies \(R(X, Y )Z = 0\). Assume also that \(M\) is a Tchebychev surface in the sense of \textit{S. L. Samelson} [Arch. Ration. Mech. Anal. 114, No. 3, 237--254 (1991; Zbl 0727.53041)]. That is, that the Tchebychev vector field \(T^{\#}\) satisfies \(\hat{\nabla}_X T^{\#} = \alpha X\) where \(\alpha\) is a differentiable function on \(M\). The main theorem in this article is an explicit classification of such hypersurfaces up to centro-affine equivalence. The classification consists of three explicit canonical families which coincide with the lists found in the related problems of \textit{H. Trabelsi} [Result. Math. 51, No. 1--2, 127--140 (2007; Zbl 1147.53010)] and \textit{A. Li} and \textit{C. Wang} [Result. Math. 20, No. 3--4, 660--681 (1991; Zbl 0752.53010)].
scientific article; zbMATH DE number 7369083