arXiv research

Rigidity of Fubini-Study metric on odd complex Grassmannians.

Study on complex Grassmannians' rigidity using Einstein deformations.

New machine learning approach finds Kähler metrics through Grassmannian learning.

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…

Harmonic unit normal sections studied for Grassmannians induced by cross products.

Study shows unique Einstein metrics on $SU_{2n+1}$ and related spaces.

Overcomplete representations and dictionary learning algorithms kept attracting a growing interest in the machine learning community. This paper addresses the emerging problem of comparing multivariate overcomplete representations. Despite a recurrent need to rely on a distance for learning or assessing multivariate ov…

The abstract introduces a new concept called flagfolds to model multi-dimensional shapes.

The period map for 4-manifolds is dense and surjective under certain conditions.

Classifies linear embeddings of grassmannians and ind-grassmannians.

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

A Riemannian manifold is said to be almost positively curved if the sets of points for which all $2$-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$ admits a metric of almost positive curvature, giving the first example of an almost posi…

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $Ω^{\otimes k}_X$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding B…

Let $A$ be a positive injective operator in a Hilbert space (\h, <,>), and denote by [,] the inner product defined by A: [f,g]=<Af,g>. A closed subspace $\s \subset \h$ is called A-compatible if there exists a closed complement for $\s$, which is orthogonal to $\s$ with respect to the inner product [,]. Equivalently, i…

Using a stability criterion due to Kröncke, we show, providing ${n\neq 2k}$, the Kähler--Einstein metric on the Grassmannian $Gr_{k}(\mathbb{C}^{n})$ of complex $k$-planes in an $n$-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Krönck…

Study proves existence of precotangent bundles for Grassmannians.

The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…

The paper finds inequalities in Grassmannian geometry.

Optimal transport theory applied to quantum states on Grassmannians.

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

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

Study second order integrability of Einstein deformations on Riemannian and Kähler 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.

In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2, published by Chen and Nagano (B.…

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.

We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\mathrm{Gr}(2, 5)\subset\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variet…

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.

Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to …

We construct a new family of compact orbifolds with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another known family, these examples classify all 4-dimensional orbifolds that are quaternion Kaehler quotients by a torus of real Grassmannians.

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…

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain dir…