arXiv research

Study identifies subvarieties of projective varieties mapping to models.

The paper classifies minimal two-spheres in quaternionic projective space.

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are …

Researchers use symplectic Dirac operator to study projective structure and its symmetries.

Characterizes the largest two-systole in real projective spaces.

Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.

We show a correspondence between the set of all G-invariant projectively flat connections on a homogeneous apace $M=G/K$, and the one of all {G}^~-invariant flat connections on a homogeneous space {M}^~={G}^~/K, where {G}^~ is a central extension of G.

New insights into continual learning for deep models, showing convergence issues but local linear solutions.

The paper proves a theorem for non-integrable projective distributions of degree one.

We classify isometric actions of compact Lie groups on quaternionic-Kähler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic.

The study classifies special manifolds leading to quaternionic Kähler manifolds.

The paper defines and proves rigidity for a special type of map between complex manifolds.

We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential varieties of homogeneous varieties already constructed. Our algorithms make no referenc…

Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.

Study of flat Legendre transforms on complex projective plane.

The study explores Einstein Kropina metrics on Lie groups and homogeneous spaces.

Study K-theory of homogeneous spaces of Lie groups.

The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.

Ricci flow preserves positive sectional curvature on homogeneous spheres

5 homogeneous convex foliations of degree 4 found on complex projective plane.

Study conjugate points on generalized flag manifolds using geodesic energy analysis.

Study $π$-equigeodesic vectors in homogeneous fibrations.

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let $\nabla$ be …

Let $M_1$ and $M_2$ be two Kähler manifolds. One says that $M_1$ and $M_2$ are "relatives" if they share a non-trivial Kähler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $h_1: S -> M_1$ and $h_2: S -> M_2$. In this paper we show that a bounded homogeneous domain …

The paper explores connections between projective Hessian and Sasakian manifolds.

Classifies star products on Lie algebroid duals and extends to projectable quantizations.

In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least $C^1$ boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…

We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…

The paper finds a correspondence between invariant PDEs and hypersurfaces.

Simpler method derived for path geometries on surfaces, characterizing projective path geometries.

Geometric structures on special manifolds are studied for rigidity.

The classification of even-homogeneous complex supermanifolds of dimension 1|m, m\leq 3, on CP^1 up to isomorphism is given. An explicit description of such supermanifolds in terms of local charts and coordinates is obtained.

We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study …

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form $μω$, admits a holomorphic isometric immersion in the complex projective space, for a suitable $μ…

New proof classifies homogeneous 3-Sasakian and quaternionic Kähler manifolds.

The paper analyzes systoles of complex projective spaces under various metrics.

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.

We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low dimensional example that can be obtained in this way.

Study proves projective Anosov subgroups lead to mixing flows in specific spaces.

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear …

Classifies special quartic curves up to equivalence.

The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.

We classify six-dimensional homogeneous nearly Kähler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly Kähler manifold is a Riemannian 3-symmetric space equipped with its canonical almost Hermitian structure. The only four examples in dimension 6 are $S^3 \times S^3$, the com…

These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. Recent results are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of t…

14 homogeneous convex foliations of degree 5 found on complex projective plane.

A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…

Analyzing Ricci Flow on 5D homogeneous manifolds.