arXiv research

The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.

Generalizes embedding complex Grassmannians into quadrics.

Classifies linear embeddings of grassmannians and ind-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.

Constructs a Morse-Bott function on symplectic Grassmannians.

Constructs explicit harmonic functions and morphisms on complex and quaternionic Grassmannians.

We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.

In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb v…

Study on complex Grassmannians' rigidity using Einstein deformations.

Study presents a twistor correspondence for specific geometric structures.

Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.

In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $δ$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hol…

Constructs explicit $p$-harmonic functions on Grassmannians and flag manifolds.

The degree of certain holomorphic 2-spheres is bounded.

The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassm…

The Grassmannian $V_2(\mathbb{R}^{n+2})$ of oriented 2-planes in $\mathbb R^{n+2}$ where $n\ge3$ carries a homogeneous parabolic contact structure of Grassmannian type. The main result of this article is that on $V_2(\mathbb{R}^{n+2})$ lives an elliptic complex of invariant differential operators of length 3 which star…

The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.

Study holomorphic isometric embeddings of a Grassmannian into quadrics.

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξφT=TR_ξφ$, where $T=A$ or $T=S$ for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Ricci tensor is parallel with respect to the generalized Tanaka-Webster connection.

Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.

The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.

We prove an extension of a theorem of A.Ros on a characterization of seven compact Kaehler submanifolds by holomorphic pinching to certain submanifolds of the complex Grassmannian manifolds.

Rigidity of Fubini-Study metric on odd complex Grassmannians.

In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $ξ$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb H}H^n$, $m=2n$, in …

We present an explicit description of all harmonic maps of finite uniton number from a Riemann surface into a complex Grassmannian. Namely, starting from a constant map $Q$ and a collection of meromorphic functions and their derivatives, we show how to algebraically construct all harmonic maps from the two-sphere into …

Study automorphisms and real structures on a special super-Grassmannian.

We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\OO\cong \C^8$. By studying the torus fixed points, we compute the Poincaré…

Explains quantum cohomology of Grassmannians using tt* equations.

In this paper we give a unified framework for the construction of complex valued harmonic morphisms from the real, complex and quaternionic Grassmannians and their non-compact duals. This gives a positive answer to the corresponding open existence problem in the real and quaternionic cases.

We use the Cartan representations of $SO(3)$ and $SU(3)$, and an irreducible 14-dimensional representation of $Sp(3)$ to construct certain totally geodesic submanifolds in "skew" position in the complex quadrics, the complex 2-Grassmannians and the quaternionic 2-Grassmannians.

Develops a correspondence between symplectic orbits and Grassmannians.

We study holomorphic extensions of Matsuki orbits in complex Grassmannians.

In this paper, we have introduced a new notion of generalized Tanaka-Webster Reeb recurrent Ricci tensor in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Next, we give a non-existence property for real hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with such a condition.

In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$.

Researchers classify special curved spheres in a complex space.

Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.

We define a torus action on the (complex) Cayley Grassmannian $X$. Using this action, we prove that $X$ is a singular variety. We also show that the singular locus is smooth and has the same cohomology ring as that of $\mathbb{CP}^5$. Furthermore, we identify the singular locus with a quotient of $G_2^\mathbb{C}$ by a …

This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a complex Grassmannian with vanishing first Chern class. The obtained manifold is a 1-torus bundle over the two dimensional real projective space. Such manifolds are interesting for mirror symmetry theory. Other examples of the s…

We give an explicit description of the non-flat parallel even Clifford structures of rank 8, 6, 5 on some real, complex and quaternionic Grassmannians, and discuss the rôle of the octonions in them, in particular for some low dimensional examples.

Let $M$ be a real hypersurface in complex Grassmannians of rank two. Denote by $\mathfrak J$ the quaternionic Kähler structure of the ambient space, $TM^\perp$ the normal bundle over $M$ and $\mathfrak D^\perp=\mathfrak JTM^\perp$. The real hypersurface $M$ is said to be $\mathfrak D^\perp$-invariant if $\mathfrak D^\p…

Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.

A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem, itself a generalization of a Theorem of Takahashi, to generalize the theory of do Carmo and Wallach and to describe the moduli…

The paper studies spectral analysis on complex spaces and finds explicit formulas for eigensections.

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new formula for the Ricci tensor of $M$ in $SU_{2,m}/S(U_2{\cdot}U_m)$. Finally we giv…

We study grassmannians associated with a linear space with a nondegenerate hermitian form. The geometry of these grassmannians allows us to explain the relation between a (pseudo-)riemannian projective geometry and the conformal structure on its ideal boundary (absolute). Such relation encompasses, for instance, the us…

The paper introduces polarizations in symplectic and orthogonal settings.

This paper connects spinors to horospheres in hyperbolic space.