Extrinsic spheres in a symmetric space
A submanifold \(N^ \ell \subset M^ n\) is called completely umbilic if \(h(X,Y) = Hg(X,Y)\) for any vector fields \(X\), \(Y\) tangent to \(N^ \ell\) where \(g\) and \(h\) are the first and second quadratic forms of \(N^ \ell\). If the mean curvature vector \(H\) is parallel to the normal connection and does not equal zero then the completely umbilic \(N^ \ell\) is called an extrinsic sphere iff it is an extrinsic hypersphere in a completely geodesic submanifold \(\widetilde{N}^{\ell + 1} \subset M^ n\) of constant curvature. The author gives the example of an extrinsic sphere in a Riemannian manifold which is not an extrinsic hypersphere in some completely geodesic submanifold.
scientific article; zbMATH DE number 484242