Some New Characterizations of Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

In this note, we establish an integral inequality for compact and orientable real hypersurfaces in complex two-plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ , involving the shape operator $A$ and the Reeb vector field $\xi$ . Moreover, this integral inequality is optimal in the sense that the real hypersurfaces attaining the equality are completely determined. As direct consequences, some new characterizations of the real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ with isometric Reeb flow can be presented.