Extrinsic hyperspheres of naturally reductive homogeneous spaces
A submanifold of a Riemannian manifold is called an extrinsic sphere if it is a totally umbilical submanifold with nonzero parallel mean curvature vector. It was proved in [\textit{B.-Y. Chen}, Mich. Math. J. 24, 265-271 (1977; Zbl 0389.53025)] that a locally symmetric space admits an \(n\)-dimensional extrinsic sphere if and only if it admits an \((n+1)\)-dimensional totally geodesic submanifold of constant sectional curvature. In this article, the author investigates a similar problem for naturally reductive homogeneous spaces and proves that if a naturally reductive homogeneous space admits an extrinsic hypersphere, then it is a space of constant sectional curvature.
scientific article; zbMATH DE number 1054588