New characterizations of real hypersurfaces with isometric Reeb flow in the complex quadric

Abstract

We prove an integral inequality for compact orientable real hypersurfaces of the complex quadric Q n (n ≥ 3) in terms of their shape operator S and Reeb vector field ξ. As direct consequences, we obtain new characterizations for real hypersurfaces of Q n with isometric Reeb flow. Such hypersurfaces have been classified by J. Berndt and Y.

Non-existence of conformally flat real hypersurfaces in both the complex quadric and the complex hyperbolic quadric

“…Moreover, in Proposition 4.1 of[3], the authors calculated the geometric invariants of all real hypersurfaces with isometric Reeb flow in the complex quadrics Q 2k (k ≥ 2), showing that each of such real hypersurfaces has at least three distinct principal curvatures. Very recently, Hu and Yin[10] obtained new characterizations of the real hypersurfaces with isometric Reeb flow in the complex quadric. Now, by Lemma 4.1, the unit normal vector field N of the conformally flat real hypersurface M is A-principal everywhere.…”