Möbius characterization of some submanifolds in the unit sphere (Q867477)

Let \(M\) be a \(m\)-dimensional umbilic-free submanifold in a \((m+p)\)-dimensional unit sphere \(S^{m+p}\). Three basic invariants of \(M\) under the Möbius transformation group of \(S^{m+p}\) are the Möbius form \(\Phi\), the Blaschke tensor \(\mathcal A\) and the Möbius metric \(g\). Denoting the Möbius scalar curvature by \(R\) and the trace-free Blaschke tensor by \({\tilde\mathcal A}=\mathcal A-\frac{1}{m}\text{ tr}({\mathcal A}) g\), the authors prove a local classification result under the assumptions of \(\Phi\equiv 0\) and the inequality \(c_1\| {\tilde \mathcal A}\| \leq R-c_2\), where \(c_1\) and \(c_2\) are some constants depending on \(m\) and \(p\).