On Conformally Flat Manifolds with Semi-parallel Ricci Tensor and Applications to the Study of Affine Hyperspheres

Abstract

This paper is concerned with the study of n-dimensional conformally flat Riemannian manifolds for \(n\ge 3\) and its applications in affine differential geometry. First, improving the work of Sekigawa–Takagi (Tohoku Math J 23:1–11, 1971), we have a complete classification for conformally flat Riemannian manifolds with semi-parallel Ricci tensor. Then, as an application, we establish a complete classification of locally strongly convex affine hyperspheres in the \((n+1)\)-dimensional affine space \({\mathbb {R}}^{n+1}\) with conformally flat affine metric and semi-parallel Ricci tensor, which generalizes the previous works of Cheng–Hu–Moruz–Vrancken (Sci China Math 63:2055–2078, 2020) and Hu–Xing (J Math Anal Appl 528:127596, 2023) on affine hyperspheres with parallel Ricci tensor.