A Möbius characterization of submanifolds (Q2501666)

In the precisely written paper the authors study Möbius characterizations of submanifolds without umbilical points in a unit sphere \(S^{n+p}(1)\). In the first part of the paper they prove that for an \(n\)-dimensional \((n\geq 2)\) submanifold \(x: M\to S^{n+p}(1)\) without umbilical points and with vanishing Möbius form \(\Phi\), if \[ (n-2)\| A\|\leq \sqrt{{n-1\over n}} \left\{nR- {1\over n}\Biggl[(n-1)\Biggl(2-{1\over p}\Biggr)- 1\Biggr]\right\} \] is satisfied, then \(x\) is Möbius equivalent to an open part of either the Riemannian product \(S^{n-1}(r)\times S^1(\sqrt{1- r^2})\) in \(S^{n+1}(1)\), or the image of the conformal diffeomorphism \(\tau\) of the Riemannian product \(S^{n-1}(x)\times H^1(\sqrt{1+ r^2})\) in \(H^{n+1}(1)\), or \(x\) is locally Möbius equivalent to the Veronese surface in \(S^4(1)\). In the second part of the work the authors consider the Möbius sectional curvature of the immersion \(x\) and obtain that, for an \(n\)-dimensional compact submanifold \(x: M\to S^{n+p}(1)\) without umbilical points and with vanishing form \(\Psi\), if the Möbius scalar curvature \(n(n-1)R\) of the immersion \(x\) is constant and the Möbius sectional curvature \(K\) of the immersion \(x\) satisfies \(K\geq 0\), when \(p= 1\) and \(K> 0\), when \(p> 1\), \(x\) is Möbius equivalent to either the Riemannian product \(S^k(r)\times S^{n-k}(\sqrt{1- r^2})\), for \(k= 1,2,\dots, n-1\), in \(S^{n+1}(1)\) or \(x\) is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in \(S^{n+p}(1)\). In this context they give interesting connections with results of M. A. Akivis and V. V. Goldberg resp. Z. J. Hu and H. Li.