Geometric and differentiable rigidity of submanifolds in spheres (Q1942685)

Let \(M\) be an \(n\)-dimensional submanifold in an \((n+p)\)-dimensional Riemannian manifold \(N\) and \(U_{x}M=\{u\in T_{x}M:\left\| u\right\| =1\}\). The authors define an extrinsic invariant \(\tau (x)\) by \[ \tau (x)=\underset{u,v\in U_{x}M\text{, }u\perp v}{\max }\left\| h(u,u)-h(v,v)\right\| ^{2}, \] where \(h\) is the second fundamental form of \(M\). Using this invariant they prove a geometric rigidity theorem for a complete submanifold with parallel mean curvature in the \((n+p)\)-dimensional unit sphere \(\mathbb{S}^{n+p}\) and a differentiable rigidity theorem for a\ complete submanifold in \(\mathbb{S}^{n+p}\). Moreover, if the ambient space is a general Riemannian manifold, they obtain a differentiable pinching theorem.