A rigidity theorem for hypersurfaces with positive Möbius Ricci curvature in \(S^{n+1}\) (Q819264)

Let \(x: M^ m\rightarrow S^ {n+1}\) be a submanifold of the unit sphere without umbilic points. Then the Möbius form \(\Phi\) and Möbius metric \(g\) of \((M^ m,x) \) are invariants of \(M^ m\) under Möbius transformations of \(S^ {n+1}\) [\textit{C. P. Wang}, Manuscr. Math. 96, 517--534 (1998; Zbl 0912.53012)]. Surfaces in \(S^ {n+1}\) with vanishing Mobius form \(\Phi\) have been classified by \textit{H. Li, C.P. Wang} and \textit{F. Wu} [Acta Math. Sin., Engl. Ser. 19, 671--678 (2003; Zbl 1078.53012), Math. Ann. 319, 707--714 (2001; Zbl 1031.53086)]. In this paper, the authors prove the following locally rigidity result for hypersurfaces: Consider an \(n\)-dimensional hypersurface \((M^ n,x)\), immersed in the unit sphere \(\mathbf{S}^ {n+1}\), \((n\geq 3)\), without umbilic points. Assume that \(\Phi =0\) and that the Ricci curvature \(\text{ Ric}_ g\) of the Mobius metric \(g\) is pinched satisfying \[ {(n-1)(n-2)\over n^ 2}\leq \text{ Ric}_ g\leq{(n^ 2-2n+5)(n-2)\over n^ 2(n-1)}. \] Then \(n=2p\), \(\text{ Ric}_ g\equiv{(n-1)(n-2)\over n^ 2}\), and \(M^ n\) is Möbius equivalent to the Einstein hypersurface \(S^ p(1/\sqrt 2)\times S^ p(1/\sqrt 2)\) of \(S^ {n+1}\).