arXiv research

Study super cluster algebras from super Plücker and Ptolemy relations.

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

A new generalization of Grassmannians, called ν-grassmannians, and a canonical super vector bundle over this new space, say Γ, are introduced. Then, constructing a Gauss supermap of a super vector bundle, the universal property of Γ is discussed. Finally, we generalize one of the main theorems of homotopy classificatio…

A new generalization of Grassmannians in supergeometry, called $ν-$Grassmannians, are constructed by gluing $ν-$domains. By a $ν-$domain, we mean a superdomain with an odd involution say $ν$ on its structure sheaf, as morphism of modules. Then we show that $ν-$Grassmannians are homogeneous superspaces. In addition, in …

We define a super analog of the classical Plücker embedding of the Grassmannian into a projective space. One of the difficulties of the problem is rooted in the fact that super exterior powers $Λ^{r|s}(V)$ are not a simple generalization from the completely even case (this works only for $r|0$ when it is possible to us…

Study non-split supermanifolds from complex manifolds.

Classifies linear embeddings of grassmannians and ind-grassmannians.

Superconformal geometries discussed in various spacetime dimensions using local supertwistor bundles.

Study proves existence of precotangent bundles for Grassmannians.

The paper finds inequalities in Grassmannian geometry.

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

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

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.

Grassmannian sigma models extend Gross-Neveu model formulations.

Correspondence found between exponential families and affine Grassmannians.

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

Differential structure on partial isometries over Grassmannian constructed.

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…

Constructs a Morse-Bott function on symplectic Grassmannians.

Generalizes embedding complex Grassmannians into quadrics.

Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.

The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $|1|$--graded parabolic geometries and for almost Grassmannian structures, in particular.…

This paper proves area-minimizing cones over Grassmannian manifolds.

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.

Develops a correspondence between symplectic orbits and Grassmannians.

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topolo…

This paper contains a thorough introduction to the basic geometric properties of the manifold of Lagrangian subspaces of a linear symplectic space, known as the Lagrangian Grassmannian. It also reviews the important relationship between hypersurfaces in the Lagrangian Grassmannian and second-order PDEs.

A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geomet…

Kernel sparsity ("dying ReLUs") and lack of diversity are commonly observed in CNN kernels, which decreases model capacity. Drawing inspiration from information theory and wireless communications, we demonstrate the intersection of coding theory and deep learning through the Grassmannian subspace packing problem in CNN…

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

Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.

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…

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…

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

Unified framework for smooth structures on coadjoint orbits.

Study on complex Grassmannians' rigidity using Einstein deformations.

Almost para-quaternionic structures on smooth manifolds of dimension $2n$ are equivalent to almost Grassmannian structures of type $(2,n)$. We remind the equivalence and exhibit some interrelations between subjects that were previously studied independently from the para-quaternionic and the Grassmannian point of view.…

Study presents a twistor correspondence for specific geometric structures.

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques d…

Study holomorphic isometric embeddings of a Grassmannian into quadrics.

In this paper we relate the geometric Poisson brackets on the Grassmannian of 2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Moebius sphere does not restrict to the space of differential invariants of Schwarzian type. But…

We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" …

Rigidity of Fubini-Study metric on odd complex Grassmannians.

Simple matrix formulas for Grassmannian curvatures.

We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e. partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group. We explain how the decomposition is concentrated in global sections for so-called (co)minuscule and (co)…

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…

The degree of certain holomorphic 2-spheres is bounded.