Poisson Quasi-Nijenhuis Manifolds

We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one correspondence with symplectic (quasi)-Nijenhuis groupoids. As an application, we study generalized complex structures in terms of Poisson quasi-Nijenhuis manifolds. We prove that a generalized complex manifold corresponds to a special class of Poisson quasi-Nijenhuis structures. As a consequence, we show that a generalized complex structure integrates to a symplectic quasi-Nijenhuis groupoid, recovering a theorem of Crainic. Francqui fellow of the Belgian American Educational Foundation. Research supported by NSF grant DMS03-06665 and NSA grant 03G-142.     

Gravitational Descendants in Symplectic Field Theory

It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems, at least in the case when the contact manifold is the prequantization space of a symplectic manifold. In this paper we generalize the definition of gravitational descendants in SFT from circle bundles in the Morse-Bott case to general contact manifolds. After we have shown using the ideas in Okounkov and Pandharipande (Ann Math 163(2):517–560, 2006) that for the basic examples of holomorphic curves in SFT, that is, branched covers of cylinders over closed Reeb orbits, the gravitational descendants have a geometric interpretation in terms of branching conditions, we follow the ideas in Cieliebak and Latschev ( [math.s6], 2007) to compute the corresponding sequence of Poisson-commuting functions when the contact manifold is the unit cotangent bundle of a Riemannian manifold.     

Locally Conformal Dirac Structures and Infinitesimal Automorphisms

We study the main properties of locally conformal Dirac bundles, which include Dirac structures on a manifold and locally conformal symplectic manifolds. It is proven that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Furthermore we show that, given a locally conformal Dirac bundle over a smooth manifold M, there is a Lie homomorphism between a subalgebra of the Lie algebra of infinitesimal automorphisms and the Lie algebra of admissible functions. We also show that Dirac manifolds can be obtained from locally conformal Dirac bundles by using an appropriate covering map. Finally, we extend locally conformal Dirac bundles to the context of Lie algebroids.     

3-Sasakian manifolds, 3-cosymplectic manifolds and Darboux theorem

We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka–Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only if it is flat.     

Coeffective and de Rham cohomologies of symplectic manifolds

We discuss the relation of the coeffective cohomology of a symplectic manifold with the topology of the manifold. A bound for the coeffective numbers is obtained. The lower bound is got for compact Kähler manifolds, and the upper one for non-compact exact symplectic manifolds. A Nomizu's type theorem for the coeffective cohomology is proved. Finally, the behaviour of the coeffective cohomology under deformations is studied.     

An integrable structure for type D spacetimes

A structure analogous to an almost complex structure on a manifold is presented. It is integrable for spacetime manifolds admitting two geodesic and shear-free null congruences. This fact sheds light on Newman's “complex coordinate transformations”.     

Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations

The leafwise complex of a reducible non-negative polarization with values in the prequantum bundle on a prequantizable symplectic manifold is studied. The cohomology groups of this complex is shown to vanish in rank less than the rank of the real part of the non-negative polarization. The Bohr-Sommerfeld set for a reducible non-negative polarization is defined. A factorization theorem is proved for these reducible non-negative polarizations. For compact symplectic manifolds, it is shown that the above complex has finite dimensional cohomology groups, more-over a Lefschetz fixed point theorem and an index theorem for these non-elliptic complexes is proved. As a corollary of the index theorem, we deduce that the cardinality of the Bohr-Sommerfeld set for any reducible real polarization on a compact symplectic manifold is determined by the volume and the dimension of the manifold. Supported in part by NSF grant DMS-93-09653, while the author was visiting University of California Berkeley.     

A remark on the contact structure on Weyl manifold and Fedosov connection

Using the contact structure on Weyl manifold, we introduce degree operator fields. The degree operator field gives a fiberwise decomposition of Weyl manifold with respect to eigenvalues. We remark that this decomposition canonically gives a symplectic connection and also a Fedosov connection on the Weyl manifold.     

A note on toric contact geometry

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective torus action whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant’s theorem due to Lerman and Tolman [E. Lerman, S. Tolman, Trans. Am. Math. Soc. 349 (10) (1997) 4201–4230] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.     

Separation of variables for integrable systems on Poisson manifolds

For an integrable system on Poisson manifolds, a construction of separated variables is discussed. We suppose that, for a given integrable system, we know a realization of the corresponding Lagrangian submanifold as the product of plane curves. In this case, we can use properties of the foliation of the initial Poisson manifold on symplectic leaves and values of the Casimir functions in order to construct separated variables.     

A Generalization of the Goldberg–Sachs theorem and its consequences

The Goldberg–Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi–Yau or symplectic and admits a solution for the source-free Einstein–Maxwell equations.     

Generalized classical BRST cohomology and reduction of Poisson manifolds

In this paper, we formulate a generalization of the classical BRST construction which applies to the case of the reduction of a Poisson manifold by a submanifold. In the case of symplectic reduction, our procedure generalizes the usual classical BRST construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold, i.e. the case of reducible first class constraints. In particular, our procedure yields a method to deal with second-class constraints. We construct the BRST complex and compute its cohomology. BRST cohomology vanishes for negative dimension and is isomorphic as a Poisson algebra to the algebra of smooth functions on the reduced Poisson manifold in zero dimension. We then show that in the general case of reduction of Poisson manifolds, BRST cohomology cannot be identified with the cohomology of vertical differential forms.Address after September 1992     

Fedosov’s formal symplectic groupoids and contravariant connections

Using Fedosov’s approach we give a geometric construction of a formal symplectic groupoid over any Poisson manifold endowed with a torsion-free Poisson contravariant connection. In the case of Kähler–Poisson manifolds this construction provides, in particular, the formal symplectic groupoids with separation of variables. We show that the dual of a semisimple Lie algebra does not admit torsion-free Poisson contravariant connections.     

Symplectic manifolds with Lagrangian fibration

A structure theorem is presented for certain kinds of symplectic manifold with a Lagrangian fibration. As a corollary, the class of cotangent bundles is characterized up to the appropriat equivalence, as the type of symplectic manifold considered in the theorem for which in addition, a certain cohomology class vanishes. These results and techniques are then applied to two situations in classical mechanics where symplectic manifolds foliated by Lagrangian submanifolds arise, namely, the Legendre transformation and Hamilton-Jacobi theory.     

A Ring-Theorist's Description of Fedosov Quantization

We present a formal, algebraic treatment of Fedosov's argument that the coordinate algebra of a symplectic manifold has a deformation quantization. His remarkable formulas are established in the context of affine symplectic algebras.     

Numerical Kähler-Einstein Metric on the Third del Pezzo

The third del Pezzo surface admits a unique Kähler-Einstein metric, which is not known in closed form. The manifold’s toric structure reduces the Einstein equation to a single Monge-Ampère equation in two real dimensions. We numerically solve this nonlinear PDE using three different algorithms, and describe the resulting metric. The first two algorithms involve simulation of Ricci flow, in complex and symplectic coordinates respectively. The third algorithm involves turning the PDE into an optimization problem on a certain space of metrics, which are symplectic analogues of the “algebraic” metrics used in numerical work on Calabi-Yau manifolds. Our algorithms should be applicable to general toric manifolds. Using our metric, we compute various geometric quantities of interest, including Laplacian eigenvalues and a harmonic (1,1)-form. The metric and (1,1)-form can be used to construct a Klebanov-Tseytlin-like supergravity solution.     

Application of the cohomology of graded Lie algebras to formal deformations of Lie Algebras

The purpose of the Letter is to show how to use the cohomology of the Nijenhuis-Richardson graded Lie algebra of a vector space to construct formal deformations of each Lie algebra structure of that space. One then shows that the de Rham cohomology of a smooth manifold produces a family of cohomology classes of the graded Lie algebra of the space of smooth functions on the manifold. One uses these classes and the general construction above to provide one-differential formal deformations of the Poisson Lie algebra of the Poisson manifolds and to classify all these deformations in the symplectic case.     

Deformations of Coisotropic Submanifolds for Fibrewise Entire Poisson Structures

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the L _∞-algebra introduced by Oh–Park (for symplectic manifolds) and Cattaneo–Felder. In the symplectic case, we recover results previously obtained by Oh–Park. Moreover we consider the extended deformation problem and prove its obstructedness.     

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.     

Constrained quantum mechanics and a coordinate independent theory of the collective path

The settings for two formulations of quantum mechanics are, respectively, Hilbert spaces and symplectic manifolds. The former leads naturally to matrix mechanics and, for example, the shell model while the latter leads to hamiltonian mechanics, of which the time-dependent Hartree-Fock theory is a standard example. In order to obtain practical approximate theories one needs to restrict the dynamics in both cases to suitable finite-dimensional subspaces. This paper addresses the problem of constructing subspaces of the projective Hilbert space, the fundamental symplectic manifold of quantum mechanics. The collective paths of Villars, Goeke and Reinhard, the valley path and the collective path and submanifold of Rowe and Basserman are examined and phrased in a coordinate independent manner. In this way we expose the dynamical foundations and the essential geometrical structures upon which they are based.