\(L_{n/2}\)-pinching theorem for submanifolds in a sphere
The author proves that for any \(n\geq 2\) there exists a constant \(C(n) >0\) such that any oriented closed \(n\)-dimensional submanifold \(M\) of the unit sphere is totally umbilical if only if the mean curvature vector of \(M\) is parallel and the second fundamental form \(A\) of \(M\) satisfies the inequality \(\int _M\| A\| ^n < C(n)\).
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