arXiv research

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

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

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…

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 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…

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

This paper proves area-minimizing cones over Grassmannian manifolds.

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…

Develops a correspondence between symplectic orbits and Grassmannians.

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

We show that there is no triangulation of the infinite real Grassmannian of k-planes in R^\infty which is nicely situated with respect to the coordinate axes. In terms of matroid theory, this says there is no triangulation of the Grassmannian subdividing the matroid stratification. This is proved by an argument in proj…

Study presents a twistor correspondence for specific geometric structures.

The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorit…

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…

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 …

The paper studies connectivity of Schur-Horn map images in real Grassmannians.

Constructs a Morse-Bott function on symplectic Grassmannians.

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 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.

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.

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…

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.

Study $Sp(n)$-orbits of isoclinic subspaces in real Grassmannians.

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…

Classifies linear embeddings of grassmannians and ind-grassmannians.

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…

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…

Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.

Using generalized Tanaka-Webster connection, we considered a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ when the GTW Reeb Lie derivative of the structure Jacobi operator coincides with the Reeb Lie derivative. Next using the method of simultaneous diagonalization, we prove a comp…

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of gene…

The paper examines when real matrix Schubert varieties are minimal submanifolds.

Study proves existence of precotangent bundles for Grassmannians.

This paper proves area-minimizing cones over products of Grassmannian manifolds.

The paper finds inequalities in Grassmannian geometry.

Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case.

The topology of the smooth moduli space of stable rank 2 bundles over a Riemann surface of genus 3 is related to that of the real Grassmannian Gr_4(R^8).

Grassmannian sigma models extend Gross-Neveu model formulations.

We classify all of real hypersurfaces $M$ with Reeb invariant shape operator in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m \geq 2$. Then it becomes a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is …

We classify, up to orbit equivalence, the cohomogeneity one actions on the noncompact duals of the symmetric spaces G_2, SU_3 and the real oriented two-plane Grassmannians.

This paper introduces a clustering framework for networks with nodes annotated with time-series data. The framework addresses all types of network-clustering problems: State clustering, node clustering within states (a.k.a. topology identification or community detection), and even subnetwork-state-sequence identificati…

Correspondence found between exponential families and affine Grassmannians.

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this correspond…

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

A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is a parallel section of the appropriate tensor bundle. We classify parallel submanifolds of the Grassmannian $\rmG^+_2(\R^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\R^{n+2}$\,. Our main resul…

The paper introduces polarizations in symplectic and orthogonal settings.

Differential structure on partial isometries over Grassmannian constructed.

The paper classifies Willmore 2-spheres in $S^n$.