Flat Hyperbolic Centro-affine Tchebychev Hypersurfaces of $\mathbb {R}^{4}$

Abstract

We study centro-affine Tchebychev hyperbolic hypersurfaces M in $\mathbb {R}^{4}$, which satisfy the following conditions: for all $X,Y,Z \in T_{p}M$:

• 1. M is flat, it means that the curvature tensor $\widehat{R}$ associated with the Levi-Civita connection of the centro-affine metric satisfies $\widehat{R}(X,Y)Z = 0$;

• 2. The Tchebychev vector field $T^{\#}$ satisfies $\widehat{\nabla }_{X}T^{\#}=\alpha X$ where $\alpha$ is a differentiable function on M, that is to say that M is a Tchebychev surface as introduced by Samelson from his work (Arch Ration Mech Anal 114:237–254) in 1991. So, we find Theorem 1 on the next page.