arXiv research

Study totally real flat minimal surfaces in hyperquadric.

The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study…

We show that non-degenerate hyperquadrics in R^{n+2} admit no skew branes. Stated more traditionally, a compact codimension-one immersed submanifold of a non-degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points. Similar results have been proven by others, but (except for e…

We classify locally the contact metric (k,mu)-spaces whose Boeckx invariant is $\le -1$ as tangent hyperquadric bundles of Lorentzian space forms.

We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric plane…

Study of minimal immersions from a sphere to a complex hyperquadric.

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices…

In this paper, we denone the generalized bicomplex numbers and give some algebraic properties of them. Also, we show that some hyperquadrics in R4 and R42 are Lie groups by using generalized bicomplex number product and obtain Lie algebras of these Lie groups. Morever, by using tensor product surfaces, we determine som…

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these function…

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further st…

In this short article, we establish a rigidity theorem for pairs of hyperquadrics in a weaker sense, i.e., we impose a condition that minimal rational curves are preserved, which is stronger than inheriting a sub-VMRT structure, a notion raised by Mok & Zhang (2014). This problem has its source in a theorem of Tsai (19…

We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled, b) for a hypersurface of positive Gaussian curvature in $\mathbb{R}^{n+1}$ to be…

E. Cartan's method of moving frames is applied to 3-dimensional manifolds $M$ which are CR-embedded in 5-dimensional real hyperquadrics $Q$ in order to classify $M$ up to CR symmetries of $Q$ given by the action of one of the Lie groups $SU(3,1)$ or $SU(2,2)$. In the latter case, the CR structure of $M$ derives from a …

We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.

In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. I…

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant princi…

Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic mapping $H_0\colon U\to \bC\bP^{N_0+1}$, where $U$ is some neighborhood of $M$ in …

The theory of harmonic vector fields on Riemannian manifolds is generalised to pseudo-Riemannian manifolds. Harmonic conformal gradient fields on pseudo-Euclidean hyperquadrics are classified up to congruence, as are harmonic Killing fields on pseudo-Riemannian quadrics. A para-Kaehler twisted anti-isometry is used to …

Extends Moutard quadric concept to higher dimensions.

We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show tha…

We show that an equivariantly embedded Hermitian symmetric space in a projective space, which contains neither a projective space nor a hyperquadric as a component, is characterized by their fundamental forms as a local submanifold of the projective space. Using some invariant-theoretic properties of the fundamental fo…

In this article, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of $C^N$ of codimension d $\ge$ 1 under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the n…

Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ is studied in this paper by harmonic maps. We construct a non-homogeneous constant curved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classification theorem of linearly full conformal minimal immersions of constant curvature from $S^2$ …

We obtain upper bounds for the first Dirichlet eigenvalue of a tube around a complex submanifold $P$ of $CP^n$ which depends only on the radius of the tube, the degrees of the polynomials defining $P$ and the first eigenvalue of some model centers of the tube. The bounds are sharp on these models. Moreover, when the mo…

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds $N^n$ contained in hypersurfaces $M^{n+1}$ of the $(n+2)$-space. We give condition under which this affine focal set is a regular hypersurface and, for curves in $3$-space, we describe its stable singulariti…

In this paper we calculate the Lagrangian Floer homology $HF(L_0, L_1 : {\mathbb Z}_2)$ of a pair of real forms $(L_0,L_1)$ in a monotone Hermitian symmetric space $M$ of compact type in the case where $L_0$ is not necessarily congruent to $L_1$. In particular, we have a generalization of the Arnold-Givental inequality…

There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conform…

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-ta…

Study splitting submanifolds in specific homogeneous spaces.

The paper proves K-stability of special Gushel-Mukai manifolds.

We classify Hopf hypersurfaces of non-flat complex space forms CP^m(4) and CH^m(-4), denoted jointly by CQ^m(4c), that are of 2-type in the sense of B. Y. Chen, via the embedding into a suitable (pseudo) Euclidean space of Hermitian matrices by projection operators. This complements and extends earlier classifications …

A ${ m G}_2$-horospherical manifold is identified by its VMRT.