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. 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. 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.

我们研究\（\ mathbb {R} ^ {4} \）中的中心仿射Tchebychev双曲超曲面M ，它满足以下条件：对于T_ {p} M \中的所有\（X，Y，Z \：

1. 1。

   M是平坦的，这意味着与中心仿射度量的Levi-Civita连接相关联的曲率张量\（\ widehat {R} \）满足\（\ widehat {R}（X，Y）Z = 0 \） ;

2. 2。

   Tchebychev向量字段\（T ^ {\＃} \）满足\（\ widehat {\ nabla} _ {X} T ^ {\＃} = \ alpha X \）其中\（\ alpha \）是可微函数在M上，也就是说M是Samelson在1991年从他的工作（Arch Ration Mech Anal 114：237–254）中引入的Tchebychev曲面。因此，我们在下一页找到了定理1。