Cyclic semiparallel Ricci tensor for harmonic curvature in the complex quadric

We introduce the notion of harmonic curvature for real hypersurfaces in the complex quadric $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$. First, we prove that the unit normal vector field N on real hypersurfaces in $Q^m$ with harmonic curvature is singular, that is, $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic. Next by using a new notion of cyclic semiparallel Ricci tensor, we give a new result on real hypersurfaces with harmonic curvature and constant mean curvature in $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$.     

Pseudo‐Einstein real hypersurfaces in the complex quadric

In this article, we introduce the notion of pseudo‐Einstein real hypersurfaces in the complex quadric and give a complete classification of such hypersurfaces.     

Real hypersurfaces in the complex hyperbolic quadric with harmonic curvature

We give a classification of real hypersurfaces in the complex hyperbolic quadric ${Q^m}^{?}=S{O}_{2,m}^o/S{O}_{2}S{O}_m$ that have constant mean curvature and harmonic curvature.     

Pseudo-anti commuting Ricci tensor for real hypersurfaces in the complex hyperbolic quadric

Science China Mathematics - We introduce a new notion of pseudo-anti commuting Ricci tensor for real hypersurfaces in the noncompact complex hyperbolic quadric $Q^{m*}=SO_{2,m}^0/SO_2SO_m$ and...     

Pseudo anti‐commuting Ricci tensor and Ricci soliton real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti commuting Ricci tensor for real hypersurfaces in complex hyperbolic two‐plane Grassmannians and prove a complete classification theorem that such a hypersurface must be a tube over a totally real totally geodesic , , a horosphere whose center at the infinity is singular or an exceptional case.     

Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, we introduce the notion of Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two‐plane Grassmannians which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. By using a new method of simultaneous diagonalizations, we give a complete classification for real hypersurfaces in complex hyperbolic two‐plane Grassmannians with the Reeb parallel Ricci tensor.     

Real hypersurfaces in the complex quadric with parallel normal Jacobi operator

First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.     

Variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation of the Kähler metric. We show as application that the Soliton-Kähler-Ricci flow generated by the Soliton-Ricci flow represents a complex strictly parabolic system of the complex components of the variation of the Kähler metric.     

Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor.     

Real hypersurfaces in the complex quadric whose structure Jacobi operator is of Codazzi type

First we introduce the notion of structure Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in with structure Jacobi operator of Codazzi type.