arXiv research

Analyzes Berry phases and connection matrices on Siegel-Jacobi spaces.

We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.

We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial …

Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the Siegel-Jacobi space.

The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this paper, we discuss the theory of the geometry and the arithmetic of the Siegel-Jacobi space.

We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

We underline some differences between the geometric aspect of Berezin's approach to quantization on homogeneous Kähler manifolds and Bergman's construction for bounded domains in $\mathbb{C}^n$. We construct explicitly the Bergman representative coordinates for the Siegel-Jacobi disk $\mathcal{D}^J_1$, which is a parti…

Geodesics on extended Siegel-Jacobi upper half-plane determined.

Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.

The coherent state representation of the Jacobi group $G^J_1$ is indexed with two parameters, $μ(=\frac{1}{\hbar})$, describing the part coming from the Heisenberg group, and $k$, characterizing the positive discrete series representation of $\text{SU}(1,1)$. The Ricci form, the scalar curvature and the geodesics of th…

Extended metric defined on Siegel-Jacobi space using invariant forms.

Equations of motion for linear Hamiltonians in the real Jacobi group

This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.

In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…

Let $\mathcal{N}$ be the space of Gaussian distribution functions over $\mathbb{R}$, regarded as a 2-dimensional statistical manifold parameterized by the mean $μ$ and the deviation $σ$. In this paper we show that the tangent bundle of $\mathcal{N}$, endowed with its natural Kähler structure, is the Siegel-Jacobi space…

We show that the Wei-Norman method applied to describe the evolution on the Siegel-Jacobi disk $\mathcal{D}^J_1=\mathcal{D}_1\times\mathbb{C}^1$, where $\mathcal{D}_1$ denotes the Siegel disk, determined by a hermitian Hamiltonian linear in the generators of the Jacobi group $G^J_1$ and Berezin's scheme using coherent …

Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.

We find the homogenous Kähler isomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi domain $\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1$ as the sum of the Kähler two-form on $\mathbb{C}$ and the one on the Siegel ball $\mathcal{D}_1$. The classical motion and quantum evolution on $\mathcal{D}^J_1$…

Abstract: Study of metrics on line bundles over complex varieties.

We determine the homogeneous Kähler diffeomorphism $FC$ which expresses the Kähler two-form on the Siegel-Jacobi ball $\mc{D}^J_n=\C^n\times \mc{D}_n$ as the sum of the Kähler two-form on $\C^n$ and the one on the Siegel ball $\mc{D}_n$. The classical motion and quantum evolution on $\mc{D}^J_n$ determined by a hermiti…

Researchers explore geometric dualities in statistical manifolds.

The real Jacobi group $G^J_1(\mathbb{R})$, defined as the semi-direct product of the group ${\rm SL}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${\rm Sp}(2,\mathbb{R})$. The left-invariant one-forms on $G^J_1(\mathbb{R})$ and their dual orthogonal left-i…