Reduction and submanifolds of generalized complex manifolds

We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensor fields.     

Reduction of Courant algebroids and generalized complex structures

We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define extended actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on CP².     

Generalized Tensor Structures of Kawaguchi Spaces

We prove that, in a generalized Kawaguchi space, there exist intrinsic almost product and almost complex structures associated to a metric of this space. We derive conditions for the integrability of these structures and find compatible affine connections.     

Generalized complex and Dirac structures on homogeneous spaces

The aim of this paper is to study generalized complex geometry (Hitchin, 2002) [6] and Dirac geometry (Courant, 1990) [3], (Courant and Weinstein, 1988) [4] on homogeneous spaces. We offer a characterization of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in sln(R). For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subalgebras.     

Batalin–Vilkovisky algebra structures on Hochschild cohomology of generalized Weyl algebras

We devote to the calculation of Batalin–Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi–Yau generalized Weyl algebras. We first establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin–Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin–Vilkovisky algebra structures for them are described completely.     

Compatible Structures on Lie Algebroids and Monge-Ampère Operators

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between PN-, P Ω- and Ω N-structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampère operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.     

On generalized Calabi-Yau nilmanifolds

We consider two possible definitions generalizing the notion of Calabi-Yau manifolds, and we describe some examples of these structures. Moreover, we prove a classification theorem for four and six complex dimensional nilmanifolds admitting an invariant generalized Calabi-Yau structure.     

Almost Complex Poisson Manifolds

In this paper we consider complex Poisson manifolds and extendthe concept of complex Poisson structure, due to Lichnerowicz to themore general concept of almost complex Poisson structures. Examples ofsuch structures and the associated generalized foliation are given.Moreover, some properties of the complex symplectic structures as wellas of the holomorphic complex Poisson structures are studied.     

Generalized Big Picard Theorem for pseudo-holomorphic maps

We show how an appropriate choice for affine connections in the target manifold allows the pseudo-holomorphic curves to be realized as harmonic maps. As an application, we present a generalized Big Picard Theorem for pseudo-holomorphic maps between manifolds with almost complex structures.     

Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian manifolds.     

Structures preserved by generalized inversion and Schur complementation

In this paper we investigate the inheritance of certain structures under generalized matrix inversion. These structures contain the case of rank structures, and the case of displacement structures. We do this in an intertwined way, in the sense that we develop an argument that can be used for deriving the results for displacement structures from thoses for rank structures. We pay particular attention to the Moore-Penrose generalized inverse, showing that for the cases of most interest, the ranks of the structure satisfied by the Moore-Penrose inverse can at most double with respect to the original ranks. We consider also the case of inheritance of structure by generalized Schur complements.     

Exact solutions of the one-dimensional generalized modified complex Ginzburg–Landau equation

The one-dimensional (1D) generalized modified complex Ginzburg–Landau (MCGL) equation for the traveling wave systems is analytically studied. Exact solutions of this equation are obtained using a method which combines the Painlevé test for integrability in the formalism of Weiss–Tabor–Carnevale and Hirota technique of bilinearization. We show that pulses, fronts, periodic unbounded waves, sources, sinks and solution as collision between two fronts are the important coherent structures that organize much of the dynamical properties of these traveling wave systems. The degeneracies of the 1D generalized MCGL equation are examined as well as several of their solutions. These degeneracies include two important equations: the 1D generalized modified Schrödinger equation and the 1D generalized real modified Ginzburg–Landau equation. We obtain that the one parameter family of traveling localized source solutions called “Nozaki–Bekki holes” become a subfamily of the dark soliton solutions in the 1D generalized modified Schrödinger limit.     

Adaptive linear generalized synchronization between two nonidentical networks

In this paper, the linear generalized synchronization between two nonidentical complex dynamical networks is investigated. Both non-delay and delay-coupled complex dynamical networks are studied. By designing effective adaptive controllers, the linear generalized synchronization between two networks with identical and nonidentical topological structures can realize. The feasibility of the proposed scheme is proved in theory and illustrative examples are presented to demonstrate the application of the theoretical results.     

Reduction of Jacobi manifolds via Dirac structures theory

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold (M,Λ,E) for which 1 is an admissible function and Jacobi quotient manifolds of M. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.     

Some generalizations of the Riemann spaces of Einstein

We introduce a class of Riemann structures, called generalized Einstein structures of index 2e, of which Einstein spaces are a particular case. We show that these structures are stationary for functions introduced on a family of Riemann structures of the compact manifold of H. Weyl. This result solves a problem of M. Berger. As examples of structures which are generalized Einstein structures over all indices we cite homogeneous compact Riemann spaces with a nondecomposable isotropy group and products of such spaces.     

Dirac structures and paracomplex manifolds

We introduce the notion of a generalized paracomplex structure. This is a natural notion which unifies several geometric structures such as symplectic forms, paracomplex structures, and Poisson structures. We show that generalized paracomplex structures are in one-to-one correspondence with pairs of transversal Dirac structures on a smooth manifold. To cite this article: A. Wade, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Weak structures

We define weak structures and show that these structures can replace in many situations generalized topologies or minimal structures.     

Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz–Schur forms

We introduce new expressions for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra under some conditions. We generalized some recent results for Drazin inverse and group inverse of complex matrices.     

Definability of second order generalized quantifiers

We study second order generalized quantifiers on finite structures. One starting point of this research has been the notion of definability of Lindström quantifiers. We formulate an analogous notion for second order generalized quantifiers and study definability of second order generalized quantifiers in terms of Lindström quantifiers.