arXiv research

The paper studies HYM connections on stable vector bundles over Kähler manifolds.

Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.

Study Spin(7) instantons and HYM connections on Stenzel metric.

We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…

For any locally defined $G_{2}-$structure and Hermitian-Yang-Mills connection on $S^{6}$, the $G_{2}-$monopole equation always admits a local solution that is asymptotic to the HYM-connection.

The paper studies connections on stable bundles and their continuity under metric variations.

This is the first nontrivial construction to date of instantons over a compact manifold with holonomy exactly $G_2$. The HYM connections on asymptotically stable bundles over Kovalev's noncompact Calabi-Yau 3-folds, obtained in the first article, are glued compatibly with a twisted connected sum, to produce a $G_2-$ins…

This paper begins to study the limiting behavior of a family of Hermitian Yang-Mills (HYM for brevity) metrics on a class of rank two slope stable vector bundles over a product of two elliptic curves with Kähler metrics $ω_ε$ when $ε\to 0$. Here $ω_ε$ are flat and have areas $ε$ and $ε^{-1}$ on the two elliptic curves …

Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid H…

We study a natural contact instanton (CI) equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (tHYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreduci…

Researchers classify and construct various instantons and connections on a specific 8-manifold.

The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.

New exponential decay estimate for Hermitian Yang-Mills metrics near branch points.

Let $Γ$ be a finite group acting linearly on $\C^n$, freely outside the origin, and let $N$ be the number of conjugacy classes of $Γ$ minus one. A construction of Kronheimer of moduli spaces $X_ζ$ of translation-invariant $Γ$-equivariant instantons on $\C^2$ is generalised to $\C^n$. The moduli spaces $X_ζ$ depend on a…

A new connection in Finsler geometry unifies various types of connections.

A $(J^{2}=\pm 1)$-metric manifold has an almost complex or almost product structure $J$ and a compatible metric $g$. We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to $J$. This projection sends the Le…

We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

New normalization condition for sub-Riemannian connections.

Odd connections on supermanifolds are defined and their properties studied.

The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.

The paper proves monotonicity formulas for minimal connections and their applications.

The paper solves the Integration Problem for principal connections.

This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case …

Study of multiplicative connections in Lie groupoids.

In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connect…

Develops torsion dual connections for statistical manifolds.

Extends connections on Lie groupoids, proving completeness conditions.

Study on solvable Lie groups with specific Weyl connections.

The study connects conic connections and torsion-free principal connections on G-structures.

Defines semi-symmetric metric connections on differential forms.

Sprays on Frechet manifolds connect connections and tangent structures.

New connections found with specific torsion properties.

Defines a new natural connection on Riemannian Π-manifolds.

In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

Paper extends Simons theorem to $F$-Yang-Mills connections for instability.

Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…

The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.

Flat Yang-Mills connections on pinched manifolds.

The study classifies holomorphic projective connections on complex threefolds.

In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…

We assume a vector bundle $p: E\to M$ with a general linear connection $K$ and a classical linear connection $\Lam$ on $M$. We prove that all classical linear connections on the total space $E$ naturally given by $(\Lam, K)$ form a 15-parameter family. Further we prove that all connections on $J^1 E$ naturally given by…

Classifies Ricci solitons on specific Lorentzian Lie groups.

Paper develops a unified framework for Lie algebroid connections on various bundles.

Characterizes connections on multivariate normal distributions.

The paper studies a new connection on Riemannian manifolds and finds conditions for symplectic manifolds.