arXiv research

Classifies special hypersurfaces in specific types of manifolds.

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

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.

A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If…

Counterexample disproves conjecture about Fano varieties with non-reductive automorphisms.

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue…

Develops a new framework for generalized Ricci flow on Lie groups.

We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the equivalence relation defined by any Polish action.

Study calculates Ricci bounds for special Fano manifolds.

A subalgebra of a Lie algebra $\mathfrak{h}\subset\mathfrak{g}$ determines $\mathfrak{h}$-representation $ρ$ on $\mathfrak{m}=\mathfrak{g}/\mathfrak{h}$. In this note we discuss how to reconstruct $\mathfrak{g}$ from $(\mathfrak{h},\mathfrak{m},ρ)$. In other words, we find all the ingredients for building non-reductive…

New representation of curves helps prove complex geometry result.

Develops a new geometric framework for quantum metrics.

For two positive integers m and n, we let ${\mathcal P}_n$ be the open convex cone in ${\mathbb R}^{n(n+1)/2}$ consisting of positive definite n x n real symmetric matrices and let ${\mathbb R}^{(m,n)}$ be the set of all m x n real matrices. In this article, we investigate differential operators on the non-reductive ma…

We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds. We show how to express this as a reductive quotient and so a GIT…

Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.

Introduces new types of homogeneous spaces and their properties.

The study finds homogeneous geodesics in homogeneous Kropina spaces.

Study finds six homogeneous surfaces with multiple invariant connections.

Introduces homogeneity supermanifolds for studying graded structures.

Homogeneous three-spheres have only homogenous foliations.

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogen…

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…

We prove that under some purely algebraic conditions every locally homogeneous structure modelled on some homogeneous space is induced by a locally homogeneous structure modelled on a different homogeneous space.

Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.

We show that a Lorentzian homogeneous space admitting a homogeneous structure of type T1 + T3 is either a (locally) symmetric space or a singular homogeneous plane wave.

In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present…

The study verifies a conjecture about homogeneous quotients of manifolds with positive curvature.

In this article we study homogeneous warped product Einstein metrics and its connections with homogeneous Ricci solitons. We show that homogeneous $(λ,n+m)$-Einstein manifolds (which are the bases of homogeneous warped product Einstein metrics) are one-dimensional extensions of algebraic solitons. This answers a questi…

The Homogeneity Conjecture explores if constant displacement isometries imply homogeneous spaces.

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …

Survey of recent results on homogeneous finite-dimensional spaces.

The paper extends two-step homogeneous geodesics to homogeneous Finsler spaces.

Smooth manifolds from locally homogeneous spaces.

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members…

Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.

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 …

We extend the definition of curvature homogeneity of type (1,3) to include the possibility that there is a homothety between any two points of a manifold preserving the first r covariant derivatives of the curvature operator simultaneously; we call this strong curvature homogeneity of type (1,3) up to order r. We chara…

The paper classifies compact homogeneous Finsler manifolds with positive flag curvature.

The paper studies Randers and $(α,β)$ equigeodesics on compact homogeneous manifolds.

Researchers found all homogeneous structure tensors on two specific 3D manifolds.

Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.

Study how regularization and optimization affect margin in deep models.

New tools compute index of symmetry in homogeneous fibrations.

Study on homogeneous geodesics in sub-Riemannian geometry.

Study shows magnetic trajectories in Berger spheres are homogeneous.

Classifies special homogeneous curves with polynomial equations.