On conformally flat manifolds with semi-parallel Ricci tensor and applications to the study of affine hyperspheres (Q6609573)

Related to a complete classification of locally strongly convex affine hyperspheres with constant sectional curvature due to \textit{L. Vrancken} et al. [Math. Z. 206, No. 4, 651--658 (1991; Zbl 0721.53014)], it is a natural and interesting problem to classify all locally strongly convex affine hyperspheres of the affine space \(\mathbb R^{n+1}\) with affine metrics being conformally flat for \(n\geq 3\). The authors study such affine hyperspheres, and classify conformally flat Riemannian manifolds with semi-parallel Ricci tensor, which improves the result due to \textit{K. Sekigawa} and \textit{H. Takagi} [Tôhoku Math. J. (2) 23, 1--11 (1971; Zbl 0218.53056)], by removing the assumption of completeness. Moreover, as an application, they establish a complete classification of locally strongly convex affine hyperspheres with conformally flat affine metric and semi-parallel Ricci tensor, which generalizes the results due to \textit{X. Cheng} et al. [Sci. China, Math. 63, No. 10, 2055--2078 (2020; Zbl 1467.53011)] and \textit{Z. Hu} and \textit{C. Xing} [J. Math. Anal. Appl. 528, No. 1, Article ID 127596, 11 p. (2023; Zbl 1539.53013)] on affine hyperspheres with parallel Ricci tensor.