arXiv subjects

Fani Petalidou

Publications and source records attributed to Fani Petalidou.

9 recordsLinked to original sources

arXiv2026

Loday algebroids cohomology, nonlinear connections and characteristic classes

The purpose of this paper is to contribute to the further study of Loday algebroids by introducing, first, the concept of action Loday algebroid and clarifying the notion of a (co)morphism of Loday algebroids. We then focus on the study of Loday algebroids nonlinear connections and their related theory of characteristic classes. These classes arise from multidifferential operators (on the module of smooth sections of the Loday algebroid) in the Loday algebroid cohomology. In particular, the Chern-Simons differential operators for nonlinear connections on Loday algebroids are considered and a generalized Chern- Simons formula for such connections is established. Using representations of Loday algebroids, we define their secondary characteristic classes.

math.DG↗

arXiv2024

Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids

We construct an algebra and a complex of multidifferential operators on tensor products of a Courant algebroid E with values in the endomorphism bundle of a smooth vector bundle B, predual of E, extending the standard complex of the Courant-Dorfman algebra of E. Also, we study Dorfman connections of E on B, and show that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of them make sense in this algebra.

math.DG↗

arXiv2017

Poisson brackets with prescribed family of functions in involution

It is well known that functions in involution with respect to Poisson brackets have a privileged role in the theory of completely integrable systems. Finding functionally independent functions in involution with a given function $h$ on a Poisson manifold is a fundamental problem of this theory and is very useful for the explicit integration of the equations of motion defined by $h$. In this paper, we present our results on the study of the inverse, so to speak, problem. By developing a technique analogous to that presented in P. Damianou and F. Petalidou, Poisson brackets with prescribed Casimirs, Canad. J. Math., 2012, vol. 64, 991-1018, for the establishment of Poisson brackets with prescribed Casimir invariants, we construct an algorithm which yields Poisson brackets having a given family of functions in involution. Our approach allows us to deal with bi-Hamiltonian structures constructively and therefore allows us to also deal with the completely integrable systems that arise in such a framework.

math.DG↗

arXiv2015

On the symplectic realization of Poisson-Nijenhuis manifolds

We consider the problem of the symplectic realization of a Poisson-Nijenhuis manifold. By applying a new technique developed by M. Crainic and I. Marcut for the study of the above problem in the case of a Poisson manifold, we establish the existence, under a condition, of a nondegenerate Poisson-Nijenhuis structure on an open neighborhood of the zero-section of the cotangent bundle of the manifold, which symplectizes the initial structure. Additionally, we present some examples.

math.DG↗

arXiv2011

Poisson brackets with prescribed Casimirs

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.

math.DG↗

arXiv2009

On twisted contact groupoids and on integration of twisted Jacobi manifolds

We introduce the concept of twisted contact groupoids, as an extension either of contact groupoids or of twisted symplectic ones, and we discuss the integration of twisted Jacobi manifolds by twisted contact groupoids. We also investigate the very close relationships which link homogeneous twisted Poisson manifolds with twisted Jacobi manifolds and homogeneous twisted symplectic groupoids with twisted contact ones. Some examples for each structure are presented.

math.DG↗

arXiv2007

On the geometric quantization of twisted Poisson manifolds

We study the geometric quantization process for twisted Poisson manifolds. First, we introduce the notion of Lichnerowicz-twisted Poisson cohomology for twisted Poisson manifolds and we use it in order to characterize their prequantization bundles and to establish their prequantization condition. Next, we introduce a polarization and we discuss the quantization problem. In each step, several examples are presented.

math.DG↗

arXiv2005

Dirac structures for generalized Courant and Courant algebroids

We establish some fundamental relations between Dirac subbundles $L$ for the generalized Courant algebroid $(A\oplus A^{\ast}, ϕ+W)$ over a differentiable manifold $M$ and the associated Dirac subbubndles $\tilde{L}$ for the corresponding Courant algebroid $\tilde{A} \oplus \tilde{A}^{\ast}$ over $M\times \R$.

math.DG↗

arXiv2004

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.

math.SG↗