arXiv research

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 construct, for a homogeneous Lagrangian of arbitrary order in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the `fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of s…

We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger's fundamental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated l…

The fundamental 2-form of an invariant almost Hermitian structure on a 6-dimensional Lie group is described in terms of an action by SO(4)xU(1) on complex projective 3-space. This leads to a combinatorial description of the classes of almost Hermitian structures on the Iwasawa and other nilmanifolds.

Study shows finiteness of magnetic hypersurfaces on closed manifolds.

We construct, for a second-order homogeneous Lagrangian in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the 'fundamental Lepage equivalent' associated with first-order Lagrangians defined on jets of section…

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.

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…

We study a class of Hermitian metrics on complex manifolds, recently introduced by J. Fu, Z. Wang and D. Wu, which are a generalization of Gauduchon metrics. This class includes the one of Hermitian metrics for which the associated fundamental 2-form is $\partial \bar\partial$-closed. Examples are given on nilmanifolds…

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

Classifies all toric Kahler surfaces with twistor 2-forms.

In recent work N. Hitchin introduced the concept of "generalised geometry". The key feature of generalised structures is that that they can be acted on by both diffeomorphisms and 2-forms, the so-called $B$-fields. In this lecture, we give a basic introduction and explain some of the fundamental ideas. Further, we disc…

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…

This paper classifies Hermitian structures on 6-dimensional nilmanifolds M=G/L for which the fundamental 2-form is d d-bar closed, a condition that is shown to depend only on the underlying complex structure J of M. The space of such J is described when G is the complex Heisenberg group, and explicit solutions are obta…

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…

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

New metrics link Kähler and pp-wave spacetimes.

We classify, up to a local isometry, all non-Kahler almost Kahler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost Kahler 4-manifolds satisfy the third curvature condition of A. Gr…

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

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…

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

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.…

Smooth low-regular connections lead to smooth immersions with controlled regularity.

Let $M$ be a compact Riemannian manifold, $π:\widetilde{M}\rightarrow M$ be the universal covering and $ω$ be a smooth $2$-form on $M$ with $π^*ω$ cohomologous to zero. Suppose the fundamental group $π_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $…

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…

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

A generalized Lepage form for second-order Lagrangians is described.

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

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

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…

A new method defines bounded cohomology classes from differential forms.

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 author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negat…

This paper is concerned with symmetries of closed multiplicative 2-forms on Lie groupoids and their infinitesimal counterparts. We use them to study Lie group actions on Dirac manifolds by Dirac diffeomorphisms and their lifts to presymplectic groupoids, building on recent work of Fernandes-Ortega-Ratiu \cite{FOR} on P…

Study Minkowski formula for conformal Killing-Yano 2-forms in constant curvature spacetimes.

We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, …

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…

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has a non-trivial, closed, self-dual 2-form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the la…

We show that the generalized Kähler-Ricci soliton equation on 4-dimensional toric Kähler orbifolds reduces to ODEs assuming there is a Hamiltonian 2-form. This leads to an explicit resolution of this equation on labeled triangles and convex labeled quadrilaterals. In particular, we give the explicit expression of the K…

We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^…

We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.

The mobility of a Kaehler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kaehler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kaehler metric to have m…

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^{…

Study symplectic structures on low dimensional 2-step nilmanifolds.