A rigidity theorem for higher codimensions (Q1067219)

Let \(f: M^ n\to R^{n+k}\) be an isometric immersion of an n- dimensional, connected, Riemannian manifold \(M^ n\) into the \((n+k)\)- Euclidean space \(R^{n+k}\), and let \(B=\{p\in M\); \(\nu (p)=n\}\) be the set of totally geodesic points of f; here \(\nu\) is the nullity of the second fundamental form of f. Assume that M is compact and that M-B is connected. It is known that if \(k=1\), f is rigid (unique up to rigid motions). Here the following generalization of this theorem is proved. Let \(\nu_ s(p)\), \(p\in M\), be the maximum of the nullities of the projections of the second fundamental form onto s-dimensional subspaces of the normal space at p (for \(k=1\), \(\nu_ s=\nu)\). For \(k>1\), assume, in addition to the above conditions, that \(\nu_ s(p)\leq n-(2s+1)\), \(1\leq s\leq k-1\). Then if \(k\leq 5\), f is rigid.