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.     

Generalized Killing Ricci tensor for real hypersurfaces in the complex quadric

First we introduce a new notion of generalized Killing Ricci tensor which is equivalent to the notion of cyclic parallel Ricci tensor for real hypersurfaces in the complex quadric $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$. Next, in terms of $\mathfrak{A}$-principal or $\mathfrak{A}$-isotropic unit normal vector fields we give a complete classification of real hypersurfaces in $Q^m=S{O}_{m+2}/S{O}_{m}S{O}_2$ with cyclic parallel Ricci tensor.     

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...     

Real hypersurfaces in the complex quadric with commuting Ricci tensor

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric Q^m = SO_m+2/SO_mSO₂. It is shown that the commuting Ricci tensor gives that the unit normal vector field N becomes A-principal or A-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in Q^m = SO_m+2/SO_mSO₂ with commuting Ricci tensor.     

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.     

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.     

Semi-parallel real hypersurfaces in complex two-plane Grassmannians

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.     

Recurrent real hypersurfaces in complex two-plane Grassmannians

Summary We introduce the notion of recurrent shape operator for a real hypersurface M in the complex two-plane Grassmannians G₂(C^m+2) and give a non-existence property of real hypersurfaces in G₂(C^m+2) with the recurrent shape operator.     

Local description of homogeneous real hypersurfaces of the two-dimensional complex space in terms of their normal equations

In the paper, a classification of real hypersurfaces of the space ℂ² that admit transitive actions of local Lie groups of holomorphic transformations is constructed. Any nonspherical Levi nondegenerate homogeneous surface is determined by the triple of real coefficientsN ₅₂₀ ² ,N ₄₄₀, ImN ₄₂₁ of a Moser normal equation. All such surfaces are described by several quadratic curves in the space of above coefficcients. This work was partially supported by RFBR grant 96-01-01002. Voronezh State Academy of Architecture and Civil Engineering. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 33–42, April–June, 2000. Translated by A. V. Loboda     

Conformally flat real hypersurfaces in nonflat complex planes

In this paper, we prove that there are no conformally flat real hypersurfaces in nonflat complex space forms of complex dimension two provided that the structure vector field is an eigenvector field of the Ricci operator. This extends some recent results by Cho (Conformally flat normal almost contact 3-manifolds, Honam Math. J. 38 (2016) 59–69) and Kon (3-dimensional real hypersurfaces with η-harmonic curvature, in: Hermitian–Grassmannian Submanifolds, Springer, Singapore, 2017, pp. 155–164).     

Ruled real hypersurfaces of a complex space form

The purpose of this paper is to define a ruled real hypersurface of a complex space formM _n (c), c≠0, and to give characterizations of this hypersurface by the infinitesimal affine transformation of the structure vector field induced on the hypersurface. Supported by Grant for the Institute of Mathematics, the University of Tsukuba, and TGRC-KOSEF (1993).     

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.     

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?H ⁿ ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H ₁ ²ⁿ⁺¹ .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ?H ⁿ .     

Viro theorem and topology of real and complex combinatorial hypersurfaces

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP ⁿ and are topologically “glued” out of algebraic hypersurfaces in (ℂ^*) ⁿ . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces. A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161. The work was completed during the stay of both authors at Max-Planck-Institu für Mathematik. The authors thank these funds and institutions for hospitality and financial support.     

Circular trajectories on real hypersurfaces in a nonflat complex space form

In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields.     

Commuting structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.