A rigidity theorem for hypersurfaces of the odd-dimensional unit sphere $\mathbb S^{2n+1}(1)$

We establish an optimal integral inequality for closed hypersurfaces in the odd-dimensional unit sphere $\mathbb S^{2n+1}(1)$ with vanishing Reeb function that involves the shape operator $A$ and the contact vector field $U$. The integral inequality is optimal in that all hypersurfaces attaining the equality are determined. Moreover, we obtain a new characterization for the Clifford hypersurfaces $\mathbb S^{2p+1}(r_1)\times \mathbb S^{2q+1}(r_2)$ in $\mathbb S^{2n+1}(1)$ with $p+q=n-1$ and $r_1^2+r_2^2=1$.