Hopf Hypersurfaces in Complex Two-Plane Grassmannians with Reeb Parallel Shape Operator

Abstract

In this paper, we consider a new notion of Reeb parallel shape operator for real hypersurfaces $$M$$ in complex two-plane Grassmannians $$G_2({\mathbb C}^{m+2})$$ . When $$M$$ has Reeb parallel shape operator and non-vanishing geodesic Reeb flow, it becomes a real hypersurface of Type $$(A)$$ with exactly four distinct constant principal curvatures. Moreover, if $$M$$ has vanishing geodesic Reeb flow and Reeb parallel shape operator, then $$M$$ is model space of Type $$(A)$$ with the radius $$r = \frac{\pi }{4\sqrt{2}}$$ .