Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel shape operator

Real hypersurfaces in the complex hyperbolic quadric with Reeb invariant Ricci tensor

“…When we consider a hypersurface M in the complex hyperbolic quadric Q m * , under the assumption of some geometric properties the unit normal vector field N of M in Q m can be divided into two classes if either N is A-isotropic or A-principal (see [3], [30], [33], and [34]). In the first case where [33] has shown that a real hypersurface M in Q m * with isometric Reeb flow is locally congruent to a tube over a totally geodesic CH k in Q 2k or a horosphere with A-isotropic center at the infinity.…”

Real hypersurfaces in the complex hyperbolic quadric with harmonic curvature

“…At each point we define which is the maximal ‐invariant subspace of . Then by using the same method for real hypersurfaces in complex hyperbolic quadric as in Suh [36], and Suh and Hwang [39] we get the following: Lemma Let M be a real hypersurface in complex hyperbolic quadric . Then the following statements are equivalent: (i)The normal vector of M is ‐principal. (ii). (iii)There exists a real structure such that . …”

“…When we consider a hypersurface M in the complex hyperbolic quadric , the unit normal vector field N of M in can be divided into two classes if either N is ‐isotropic or ‐principal (see [33], [34], [36] and [39]). In the first case where N is ‐isotropic, we have shown in Theorem 1.2 that M is locally congruent to a tube over a totally geodesic complex hyperbolic space in or a horosphere.…”

“…Recently we have given a complete classification on real hypersurfaces in Hermitian symmetric spaces with isometric Reeb flow (see Berndt and Suh [6]). In particular, compared to real hypersurfaces in the complex quadric due to Berndt and Suh [4], and Suh [34], [38] with isometric Reeb flow , parallel Ricci tensor and Reeb parallel Ricci tensor , we have focused our study on the parallelism of shape operator and Ricci tensor in the complex hyperbolic quadric given in Klein and Suh [14], Kim and Suh [10], and Suh and Hwang [39] respectively.…”

Generalized Killing Ricci tensor for real hypersurfaces in the complex quadric

“…Now we consider the Ricci operator $$Ric$$ on a real hypersurface M in the complex quadric $$Q^m$$. When we consider a hypersurface M in the complex quadric $${Q}^m$$, the unit normal vector field N of M in $$Q^m$$ can be divided into two classes according to N is $$\mathfrak {A}$$‐isotropic or $$\mathfrak {A}$$‐principal (see [27, 29, 31, 33, 34]). A real hypersurface M in $$Q^m$$ is said to be isometric Reeb flow if the structure tensor ϕ commutes with the shape operator S of M in $$Q^m$$.…”