The classifications of locally strongly convex equiaffine hypersurfaces (resp. centroaffine hypersurfaces) with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Blaschke–Berwald affine metric (resp. centroaffine metric) have been completed in the last decades. In this paper we define generalized Calabi products in Calabi geometry and prove decomposition theorems in terms of their Calabi invariants. As the main result, we obtain a complete classification of Calabi hypersurfaces in \({{\mathbb {R}}}^{n+1}\) with parallel Fubini–Pick tensor with respect to the Levi-Civita connection of the Calabi metric. This result is a counterpart in Calabi geometry of the classification theorems in equiaffine situation Hu et al. (J Diff Geom 87:239-307, 2011) and centroaffine situation Cheng et al. (Results Math 72:419-469, 2017).

具有平行 Fubini-Pick 张量的局部强凸等仿射超曲面（或中心仿射超曲面）相对于 Blaschke-Berwald 仿射度量（或中心仿射度量）的 Levi-Civita 连接的分类已在过去几十年中完成。在本文中，我们定义了卡拉比几何中的广义卡拉比积，并根据其卡拉比不变量证明了分解定理。作为主要结果，我们获得了\({{\mathbb {R}}}^{n+1}\)中 Calabi 超曲面的完整分类，其中具有关于 Calabi 的 Levi-Civita 连接的并行 Fubini-Pick 张量公制。这个结果是等仿射情况下分类定理的 Calabi 几何的对应结果 Hu 等人。 (J Diff Geom 87:239-307, 2011) 和中心仿射情况 Cheng 等人。 （数学结果 72：419-469，2017 年）。