arXiv research

Study invariant CKY 2-forms on 5D Lie groups, classifying and determining their properties.

Study magnetic fields on special Lie groups, proving non-existence of certain types.

Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kähler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for a…

The paper studies symplectic structures on character varieties of Sasakian threefolds.

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half conformally flat. In the first case, the problem reduces t…

This paper explains a 2-form on moduli spaces using quasi-Hamiltonian and Dirac geometry.

Study of presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.

In this paper, M denotes a smooth manifold of dimension n, A a Weil algebra and M^{A} the associated Weil bundle. When (M,_{M}) is a Poisson manifold with 2-form _{M}, we construct the 2-Poisson form _{M^{A}}^{A}, prolongation on M^{A} of the 2-Poisson form _{M}. We give a necessary and sufficent condition for that M^{…

The paper generalizes Segre and Verlinde numbers for surfaces with holomorphic 2-forms.

The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of seq…

The paper extends the classification of bi-invariant 2-forms to infinite-dimensional Lie groups.

Maps Lie 2-groups to Weil algebras, showing cohomology isomorphisms.

We introduce coG_2-vector fields, coRochesterian 2-forms and coRochesterian vector fields on manifolds with a coclosed G_2-structure as a continuous of work from [15], and we show that the spaces of coG_2-vector fields and of coRochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the …

A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form $ω$ is $\partial \bar \partial$-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a $2n$-dimensional SKT Lie algebra $\mathfr…

We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poiss…

The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details -- in particular the global moduli / instanton / magnetic charge structure -- have remained elusive. Here we deduce from…

This note is devoted to partial study of recurrent equation $dω=β\wedge ω$, based on linear algebra of exterior forms. Such equation was considered by Lee, for non-degenerate 2-form. In this note we approach general case, when $ω$ is arbitrary. Particularly, we extend results obtained by Lee, on odd-forms.

We consider a vector space V over K=R or C, equipped with a skew symmetric bracket [.,.]: V x V --> V and a 2-form omega:V x V --> K. A simple change of the Jacobi identity to the form [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=omega(B,C)A+omega(A,B)C+omega(C,A)B opens new possibilities, which shed new light on the Bianchi classifi…

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the symplectic diffeomorphisms group $\mathcal{D}_ω$ of a symplectic Riemannian manifold $(M,g,ω)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_ω$ and calculate the sectional curvature of …

Study shows exact forms in bounded cohomology are in radical of cup product.

We study the relationship between multiplicative 2-forms on Lie groupoids and linear 2-forms on Lie algebroids, which leads to a new approach to the infinitesimal description of multiplicative 2-forms and to the integration of twisted Dirac manifolds.

Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved to be isomorphic to the algebra of smooth $Sp(Ω_{x})$-invariant fun…

Proves magnetic geodesic flow on sphere is integrable with constant 2-form.

The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $Ω$ is 4, then the equivalence under diffeomorphisms of $Ω$ is reduced to the equivalence of a symplectic linear frame functorially attached to $Ω$. As the equi…

Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivi…

In the $3$-gauge theory, a $3$-connection is given by a $1$-form $A$ valued in the Lie algebra $\mathfrak g$, a $2$-form $B$ valued in the Lie algebra $\mathfrak h$ and a $3$-form $C$ valued in the Lie algebra $\mathfrak l$, where $(\mathfrak g,\mathfrak h, \mathfrak l)$ constitutes a differential $2$-crossed modu…

Classifies all toric Kahler surfaces with twistor 2-forms.

This paper has been withdrawn in order to replace it by two separate submissions: 1. Hamiltonian 2-forms in Kahler geometry III: Extremal metrics and stability, math.DG/0511118; 2. Hamiltonian 2-forms in Kahler geometry IV: Weakly Bochner-flat Kahler manifolds, math.DG/0511119. As the titles indicate, the first paper c…

Unified framework for Riemannian, Kahler, and hyper-Kahler geometries in 4D.

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemanni…

The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.

In this article we show that the only 2-step nilpotent Lie groups which carry a non-degenerate left invariant Killing-Yano 2-form are the complex Lie groups. In the case of 2-step nilpotent complex Lie groups arising from connected graphs, we prove that the space of left invariant Killing-Yano 2-forms is one-dimensiona…

Thurston introduced shear deformations (cataclysms) on geodesic laminations - deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined…

We solve higher-order morphisms for twisted Courant algebras.

Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the contact diffeomorphisms group $\mathcal{D}_θ$ of a contact Riemannian manifold $(M,g,θ)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_θ$ and calculate the sectional curvature of $\math…

Establishes a connection between Kähler metrics and vector bundle sections.

The paper classifies topological properties of specific geometric forms on manifolds.

We will prove a Moser-type theorem for self-dual harmonic 2-forms on closed 4-manifolds, and use it to classify local forms on neighborhoods of singular circles on which the 2-form vanishes. Removing neighborhoods of the circles, we obtain a symplectic manifold with contact boundary - we show that the contact form on e…

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algeb…

A 2-form on a quaternionic-Kahler manifold (M, g) is called compatible (with the quaternionic structure) if it is a section of the direct sum bundle S^2(H) \oplus S^2(E). We construct a connection D on S^2(H) \oplus S^2(E)\oplus TM, which is a prolongation of the conformal-Killing operator acting on compatible 2-forms.…

Deforms Poisson quasi-Nijenhuis manifolds using closed 2-forms.

This paper concerns the explicit construction of extremal Kaehler metrics on total spaces of projective bundles, which have been studied in many places. We present a unified approach, motivated by the theory of hamiltonian 2-forms (as introduced and studied in previous papers in the series) but this paper is largely in…

We study the construction and classification of weakly Bochner-flat (WBF) metrics (i.e., Kahler metrics with coclosed Bochner tensor) on compact complex manifolds. A Kahler metric is WBF if and only if its `normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in th…

In this paper, we investigate the action of pseudogroup of all point transformations on the natural bundle of equations $y'' = u^0(x,y) + u^1(x,y)y' + u^2(x,y)(y')^2 + u^3(x,y)(y')^3$. We calculate the 1-st nontrivial differential invariant of this action. It is a horizontal differential 2-form with values in some alge…

We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of $d ω$ and its Hodge dual $* dω$ where $ω$ is the fundamental 2-form of the nearly Kähler stru…

We study the relation between $J$-anti-invariant $2$-forms and pseudoholomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher di…

The paper generalizes the number of complex structures on metric Lie algebras.