Complex Quadrics Research Articles

New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

A real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ inherits an almost contact metric structure . This structure allows to define, for any nonnull real number $k$, the so called $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. If $\nabla$ denotes the Levi-Civita connection on $M$, we introduce the concepts of $(\hat{\nabla}^{(k)},\nabla)$-Codazzi and $(\hat{\nabla}^{(k)},\nabla)$-Killing shape operator $S$ of the real hypersurface and classify real hypersurfaces in $Q$ satisfying any of these conditions.

On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric

ABSTRACT The almost contact metric structure that we have on a real hypersurface M in the complex quadric Qm = SO m+2/SO m SO 2 allows us to define, for any nonnull real number k, the k-th generalized Tanaka-Webster connection on M, ∇ ^ ( k ) . Associated to this connection, we have Cho and torsion operators F X ( k ) and T X ( k ) , respectively, for any vector field X tangent to M. From them and for any symmetric operator B on M, we can consider two tensor fields of type (1,2) on M that we denote by B F ( k ) and B T ( k ) , respectively. We classify real hypersurfaces M in Qm for which any of those tensors identically vanishes, in the particular case of B being the structure Lie operator Lξ on M.

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

Cyclic Parallel Structure Jacobi Operator for Real Hypersurfaces in the Complex Quadric

The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric

We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor.

Curvature Invariants of Equivariant Isometric Minimal Immersions into Grassmannian Manifolds

Using theories of principal fibre bundles and connections in [11], we study curvature invariants as Riemannian submanifolds for equivariant isometric minimal immersions from Riemannian or Hermitian symmetric spaces of compact type into Grassmannian manifolds. First we calculate principal curvatures of all equivariant full isometric minimal immersions from the complex projective line to complex quadrics. It is shown that they do not depend on the parameter of the moduli space. According to the preceding research, it is important for the study of curvature invariants to examine the behavior of higher order covariant derivative of the second fundamental form. It also enables us to distinguish those maps by curvature invariants in the moduli space. In addition, we calculate principal curvatures of all equivariant full holomorphic isometric embeddings between the complex projective spaces.

Foliated Hopf hypersurfaces in complex hyperbolic quadrics

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kähler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kähler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kähler manifolds.

The $\mathfrak{A}$-principal real hypersurfaces in complex quadrics

The $\mathfrak{A}$-principal real hypersurfaces in complex quadrics

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504 ). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kahler structure with an $$S^1$$ -action on the moduli spaces of holomorphic isometric embeddings of a compact Kahler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

Ruled Real Hypersurfaces in the Complex Quadric

First we introduce the notions of $$\eta $$ -parallel and $$\eta $$ -commuting shape operator for real hypersurfaces in the complex quadric $$Q^m = SO_{m+2}/SO_mSO_2$$ . Next we give a complete classification of real hypersurfaces in the complex quadric $$Q^m$$ with such kind of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurface foliated by complex totally geodesic hyperplanes $$Q^{m-1}$$ in $$Q^m$$ whose unit normal vector field in $$Q^m$$ is $$\mathfrak {A}$$ -principal.

Mironov Lagrangian cycles in algebraic varieties

We generalize a construction due to Mironov. Some time ago he presented new examples of minimal and Hamiltonian minimal Lagrangian submanifolds in and . His construction is based on the considerations of a noncomplete toric action of , where , on subspaces that are invariant with respect to the action of a natural antiholomorphic involution. This situation takes place for a rather broad class of algebraic varieties: complex quadrics, Grassmannians, flag varieties and so on, which makes it possible to construct many new examples of Lagrangian submanifolds in these algebraic varieties. Bibliography: 4 titles.

Commuting Jacobi Operators on Real Hypersurfaces of Type B in the Complex Quadric

In this paper, first, we investigate the commuting property between the normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$ for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$, $m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$. Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi operators.

New Conditions on Normal Jacobi Operator of Real Hypersurfaces in the Complex Quadric

On a real hypersurface M of a complex quadric we have an almost contact metric structure induced by the Kahlerian structure of the ambient space. Therefore, on M we have the Levi-Civita connection $$\nabla $$ and, for any non-null real number k, the so called kth generalized Tanaka Webster connection $${\hat{\nabla }}^{(k)}$$ . We introduce the notions of $$({\hat{\nabla }}^{(k)},\nabla )$$ -Codazzi and $$({\hat{\nabla }}^{(k)},\nabla )$$ -Killing normal Jacobi operator on such a real hypersurface and classify Hopf real hypersurface in a complex quadric whose normal Jacobi operators satisfy any of both conditions.

Real Hypersurfaces of Complex Quadric in Terms of Star-Ricci Tensor

In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.

Extremities Involving B. Y. Chen’s Invariants for Real Hypersurfaces in Complex Quadric

Extremities Involving B. Y. Chen’s Invariants for Real Hypersurfaces in Complex Quadric

Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

On a real hypersurface in the complex quadric we can consider the Levi-Civita connection and, for any nonnull constant k, the k-th generalized Tanaka-Webster connection. We also have a differential operator of first order of Lie type associated to the k-th generalized Tanaka-Webster connection. We prove non-existence of real hypersurfaces in the complex quadric for which the covariant derivatives associated to both connections coincide or Lie derivative and Lie type differential operator coincide when they act on the shape operator of the real hypersurface.

Special Bohr–Sommerfeld Lagrangian submanifolds of algebraic varieties

In this paper we continue our study of special Bohr– Sommerfeld submanifolds in the case when the ambient symplectic manifold possesses a compatible integrable complex structure (and is thus an algebraic variety). In this case we show how to reduce the special Bohr– Sommerfeld geometry to Morse theory on the complements of ample divisors. This gives rise to a construction of Lagrangian shadows of ample divisors in algebraic varieties, which is an example of `algebraic v. symplectic' duality. We suggest a condition for the existence of a Lagrangian shadow and give examples of Lagrangian shadows of ample divisors on the projective plane, complex quadrics and flag manifolds.

Real Hypersurfaces with Killing Shape Operator in the Complex Quadric

We introduce the notion of Killing shape operator for real hypersurfaces in the complex quadric $$Q^m = SO_{m+2}/SO_mSO_2$$ . The Killing shape operator condition implies that the unit normal vector field N becomes $$\mathfrak {A}$$ -principal or $$\mathfrak {A}$$ -isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in $$Q^m = SO_{m+2}/SO_mSO_2$$ with Killing shape operator.

Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow

We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics [Formula: see text], [Formula: see text]. We show that [Formula: see text] is even, say [Formula: see text], and any such hypersurface becomes an open part of a tube around a [Formula: see text]-dimensional complex hyperbolic space [Formula: see text] which is embedded canonically in [Formula: see text] as a totally geodesic complex submanifold or a horosphere whose center at infinity is [Formula: see text]-isotropic singular. As a consequence of the result, we get the nonexistence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics [Formula: see text], [Formula: see text].

Pseudotoric structures on a hyperplane section of a toric manifold

We continue to investigate pseudotoric geometry. Namely, we present a type of nontoric manifolds with pseudotoric structures. As an application, we construct a new pseudotoric structure on two-dimensional complex quadrics.

REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.