A characterization of submanifolds of extrinsic symmetric spaces
A submanifold \(S\) of a Euclidean space is said to be a full extrinsically symmetric space if \(S\) is compact and the second fundamental form \(\beta^S\) of \(S\) is parallel and onto. Let \(M\) be a submanifold of \({\mathbb R}^n\) and \(S\) a full extrinsically symmetric space in \({\mathbb R}^n\). Assume that there exist a subbundle \(E \subset M \times {\mathbb R}^n\) with \(TM \subset E\) and a parallel bundle homomorphism \(\beta : E \otimes E \to E^\perp\) satisfying \(\beta(v,a) = (\partial_va)^{E^\perp}\) for all \(v \in TM\) and \(a \in E\). Assume furthermore that \(\beta\) is congruent to \(\beta^S\) at some point. The author proves that \(M\) is contained in \(S\) up to a rigid motion of \({\mathbb R}^n\). This generalizes the well-known result for the special case when \(S = S^{n-1} \subset {\mathbb R}^n\).
scientific article; zbMATH DE number 5713585