A characterization of extrinsic spheres in a Riemannian manifold

Abstract

We give a characterization of a totally umbilic submanifold $M^n$ with parallel mean curvature vector of a Riemannian manifold $\tilde{M}^{n+p}$, that is an extrinsic sphere $M^n$ of $\tilde{M}^{n+p}$, in terms of the extrinsic shape of circles on $M^n$ in the ambient manifold $\tilde{M}^{n+p}$. This characterization is an improvement of Nomizu and Yano's result ([2]).