Fluctuation of sectional curvature for closed hypersurfaces. (Q1414955)

Let \(f: M\to (\mathbb{R}^{n+1},\text{can})\) be an immersion as a hypersurface of a smooth closed manifold \(M\) and denote by \(f^*\)can the pullback metric on \(M\) of the canonical metric can on \(\mathbb{R}^{n+1}\). If \(M\) is not diffeomorphic to \(S^n\), then the sectional curvature of \(M\) with respect to this metric cannot be constant and it fluctuates inside a closed bounded interval with length \(\ell\)(sec). In this short note, the author derives a lower bound for \(\ell\)(sec) with respect to \(f^*\)can in function of the topology of \(M\) (more specifically, the sum of the Betti numbers) and the volume of \(M\) (with respect to \(f^*\)can). The obtained inequality is discussed and improved, in particular for the case of smooth, closed surfaces by using the Gauss-Bonnet theorem.