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Izu Vaisman

Publications and source records attributed to Izu Vaisman.

At least 19 recordsLinked to original sources
arXiv2017

On hypersurfaces of generalized Kähler manifolds

We establish the conditions for the induced generalized metric F structure of an oriented hypersurface of a generalized Kähler manifold to be a generalized CRFK structure. Then, we discuss a notion of generalized almost contact structure on a manifold $M$ that is suggested by the induced structure of a hypersurface. Such a structure has an associated generalized almost complex structure on $M\times\mathds{R}$. If the latter is integrable, the former is normal and we give the corresponding characterization. If the structure on $M\times\mathds{R}$ is generalized Kähler, the structure on $M$ is said to be binormal. We characterize binormality and give an example of binormal structure.

math.DG↗
arXiv2017

A note on submanifolds of generalized Kähler manifolds

In this note, we consider submanifolds of a generalized Kähler manifold that are CR-submanifolds for the two associated Hermitian structures. Then, we establish the conditions for the induced, generalized F structure to be a CRFK structure. The results extend similar conditions which we obtained for hypersurfaces in an earlier paper.

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arXiv2016

Quasi-classical generalized CRF structures

In an earlier paper, we studied manifolds $M$ endowed with a generalized F structure $Φ\in End(TM\oplus T^*M)$, skew-symmetric with respect to the pairing metric, such that $Φ^3+Φ=0$. Furthermore, if $Φ$ is integrable (in some well-defined sense), $Φ$ is a generalized CRF structure. In the present paper we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields $(A\in End(TM),π\in\wedge^2TM)$ where $A^3+A=0$ and some relations between $A$ and $π$ hold. We establish the integrability conditions in terms of $(A,π)$. They include the facts that $A$ is a classical CRF structure, $π$ is a Poisson bivector field and $im\,A$ is a (non)holonomic Poisson submanifold of $(M,π)$. We discuss the case where either $ker\,A$ or $im\,A$ is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of $im\,A$ inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of $π$, including an associated spectral sequence and a Dolbeault type grading.

math.DG↗
arXiv2015

A note on submanifolds and mappings in generalized complex geometry

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kaehler manifolds that have an induced generalized Kaehler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kaehler manifold.

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arXiv2015

Generalized para-Kähler manifolds

We define a generalized almost para-Hermitian structure to be a commuting pair $(\mathcal{F},\mathcal{J})$ of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähler structure. This class of structures contains both the classical para-Kähler structure and the classical Kähler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple $(γ,ψ,F)$, where $γ$ is a (pseudo) Riemannian metric, $ψ$ is a $2$-form and $F$ is a complex $(1,1)$-tensor field such that $F^2=Id,γ(FX,Y)+γ(X,FY)=0$. We deduce integrability conditions similar to those of the generalized Kähler structures and give several examples of generalized para-Kähler manifolds. We discuss submanifolds that bear induced para-Kähler structures and, on the other hand, we define a reduction process of para-Kähler structures.

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arXiv2013

Generalized Sasaki metrics on tangent bundles

We define a class of metrics that extend the Sasaki metric of a tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback of the big tangent bundle of a manifold to the tangent manifold. We obtain the expression of the transferred metric and we define a canonical, metric connection with torsion. We calculate the torsion, curvature and Ricci curvature of this connection and give a few applications of the results. We also discuss the transfer of generalized complex and generalized Kaehler structures.

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arXiv2013

Towards a double field theory on para-Hermitian manifolds

In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-Kähler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangent bundle of the manifold has a natural, metric-compatible bracket that extends the C-bracket of double field theory. In the para-Kähler case this bracket is equal to the sum of the Courant brackets of the two Lagrangian foliations of the manifold. Then, we define a canonical connection and an action of the field that correspond to similar objects of double field theory. Another section is devoted to the Marsden-Weinstein reduction in double field theory on para-Hermitian manifolds. Finally, we give examples of fields on some well-known para-Hermitian manifolds.

math.DG↗
arXiv2013

Geometry on Big-Tangent Manifolds

Motivated by generalized geometry, we discuss differential geometric structures on the total space $\mathfrak{T}M$ of the bundle $TM\oplus T^*M$, where $M$ is a differentiable manifold; $\mathfrak{T}M$ is called a big-tangent manifold. The vertical leaves of the bundle are para-Hermitian vector spaces. The big-tangent manifolds are endowed with canonical presymplectic, Poisson and 2-nilpotent structures. We discuss lifting processes from $M$ to $\mathfrak{T}M$. From the point of view of the theory of $G$-structures, the structure of a big-tangent manifold is equivalent with a suitable triple $(P,Q,S)$, where $P$ is a regular bivector field, $Q$ is a 2-contravariant symmetric tensor field of the same rank as $P$ and $S$ is a 2-nilpotent $(1,1)$-tensor field. The integrability conditions include the annulation of the Schouten-Nijenhuis bracket $[P,P]$, the annulation of the Nijenhuis tensor $\mathcal{N}_S$ and conditions that connect between the three tensor fields. We discuss horizontal bundles and associated linear connections with the Bott property. Then, we discuss metrics on the vertical bundle that are compatible with the para-Hermitian metric of the leaves. Together with a horizontal bundle, such metrics may be seen as a generalization of the double fields of string theory with the role of double fields over a manifold. We define a canonical connection and the action functional of such a field.

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arXiv2012

Hamiltonian vector fields on almost symplectic manifolds

Let $(M,ω)$ be an almost symplectic manifold ($ω$ is a non degenerate, not closed, 2-form). We say that a vector field $X$ of $M$ is locally Hamiltonian if $L_Xω=0,d(i(X)ω)=0$, and it is Hamiltonian if, furthermore, the 1-form $i(X)ω$ is exact. Such vector fields were considered in a 2007 paper by F. Fasso and N. Sansonetto, under the name of strongly Hamiltonian, and a corresponding action-angle theorem was proven. Almost symplectic manifolds may have few, non-zero, Hamiltonian vector fields or even none. Therefore, it is important to have examples and it is our aim to provide such examples here. We also obtain some new general results. In particular, we show that the locally Hamiltonian vector fields generate a Dirac structure on $M$ and we state a reduction theorem of the Marsden-Weinstein type. A final section is dedicated to almost symplectic structures on tangent bundles.

math.SG↗
arXiv2012

On the geometry of double field theory

Double field theory was developed by theoretical physicists as a way to encompass $T$-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms, in the framework of para-Kaehler manifolds. We define metric algebroids, which are vector bundles with a bracket of cross sections that has the same metric compatibility property as a Courant bracket. We show that a double field gives rise to two canonical connections, whose scalar curvatures can be integrated to obtain actions. Finally, in analogy with Dirac structures, we define and study para-Dirac structures on double manifolds.

math.DG↗
arXiv2011

Dirac structures on generalized Riemannian manifolds

We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.

math.DG↗
arXiv2010

Soldered tensor fields of normalized submanifolds

In an earlier paper we discussed soldered forms, multivector fields and Riemannian metrics. In particular, we showed that a Riemannian submanifold is totally geodesic iff the metric is soldered to the submanifold. In the present note we discuss general, soldered tensor fields. In particular, we prove that the almost complex structure of an almost Kähler manifold is soldered to a submanifold iff the latter is an invariant, totally geodesic submanifold.

math.DG↗
arXiv2010

From generalized Kaehler to generalized Sasakian structures

This is an expository paper, which provides a first introduction to geometric structures on $TM\oplus T^*M$. The paper contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal, almost) contact and generalized Sasakian structures. A few of these properties are new.

math.DG↗
arXiv2010

A construction of Courant algebroids on foliated manifolds

For any transversal-Courant algebroid $E$ on a foliated manifold $(M,\mathcal{F})$, and for any choice of a decomposition $TM=T\mathcal{F}\oplus Q$, we construct a Courant algebroid structure on $T\mathcal{F}\oplus T^*\mathcal{F}\oplus E$.

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arXiv2009

Foliated Lie and Courant Algebroids

If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's super-vector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid $A$ as a bracket-closed, isotropic subbundle $B$ with anchor image $T\mathcal{F}$ and such that $B^\perp/B$ is locally equivalent with Courant algebroids over the slice manifolds of $\mathcal{F}$. Examples that motivate the definition are given.

math.DG↗
arXiv2009

Conformal changes of generalized complex structures

A conformal change of $TM\oplus T^*M$ is a morphism of the form $(X,α)\mapsto(X,e^τα)$ $(X\in TM,α\in T^*M,τ\in C^\infty(M))$. We characterize the generalized almost complex and almost Hermitian structures that are locally conformal to integrable and to generalized Kähler structures, respectively, and give examples of such structures.

math.DG↗
arXiv2009

Characteristic Classes of Lie Algebroid Morphisms

We extend R. Fernandes' construction of secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism.

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