A characterization of spheres in \(\mathbb{R}{} ^ 4\) (Q1180131)

The author provides two sufficient conditions for a hypersurface in \(\mathbb{R}^ 4\) to be a sphere. Let \(M\) be a connected compact hypersurface in \(\mathbb{R}^ 4\), \(g\) the induced metric on \(M\) from the usual Euclidean metric \(\langle\;,\;\rangle\) on \(\mathbb{R}^ 4\), \(N\) a unit normal field on \(M\), \(A\) the shape operator of \(M\) with respect to \(N\), \(\alpha\) the mean curvature of \(M\), \(T\) the position vector field on \(\mathbb{R}^ 4\) restricted to \(M\), and \(J_ 1,J_ 2,J_ 3\) the tensor fields on \(\mathbb{R}^ 4\) defining its usual quaternionic structure. Then \(M\) is a sphere of a certain radius in \(\mathbb{R}^ 4\) if (1) \(M\) has positive scalar curvature, the functions \(\langle J_ iT,N\rangle\) \((i=1,2,3)\) are either non-negative or non-positive, and the function \(\langle T,N\rangle\) is nowhere zero, or if (2) there exists a parallel complex structure \(J\) on \(\mathbb{R}^ 4\) such that \(3\alpha g(AJN,JN)-\hbox{trace} (A^ 2)\geq0\).