Remarks on the geometry of the extended Siegel--Jacobi upper half-plane
Elena Mirela Babalic, Stefan Berceanu
备注：
26 pages, Latex, amsart, AMS fonts; the abstract and the introduction are improved, some typos are eliminated, more references are added; arXiv admin note: text overlap with arXiv:1903.10721
The real Jacobi group $G^J_1(\mathbb{R})={\rm SL}(2,\mathbb{R})\ltimes {\rm H}_1$, where ${\rm H}_1$ denotes the 3-dimensional Heisenberg group, is parametrized by the $S$-coordinates $(x,y,θ,p,q,κ)$. We show that the parameter $η$ that appears passing from Perelomov's un-normalized coherent state vector based on the Siegel--Jacobi disk $\mathcal{D}^J_1$ to the normalized one is $η=q+\rm{i} p$. The two-parameter invariant metric on the Siegel--Jacobi upper half-plane $\mathcal{X}^J_1=\frac{G^J_1(\R)}{\rm{SO}(2)\times\mathbb{R}}$ is expressed in the variables $(x,y,\rm{Re}~η,\rm{Im}~η)$. It is proved that the five dimensional manifold $\tilde{\mathcal{X}}^J_1=\frac{G^J_1(\R)}{\rm{SO}(2)}\approx\mathcal{X}^J_1\times\mathbb{R}$, called extended Siegel--Jacobi upper half-plane, is a reductive, non-symmetric, non-naturally reductive manifold with respect to the three-parameter metric invariant to the action of $G^J_1(\mathbb{R})$, and its geodesic vectors are determined.
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 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 partially bounded manifold whose points belong to $\mathbb{C}\times\mathcal{D}_1$, where $\mathcal{D}_1$ denotes the Siegel disk. The Bergman representative coordinates on $\mathcal{D}^J_1$ are globally defined, the Siegel-Jacobi disk is a normal Kähler homogeneous Lu Qi-Keng manifold…
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 appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, relationship to the Jacobi group), and are related to the quan…
The semidirect product of the real Heisenberg group ${\rm H}_1(\mathbb{R})$ with ${\rm SL}(2,\mathbb{R})$, called the real Jacobi group $G^J_1(\mathbb{R})$, admits a four-parameter invariant metric expressed in the S-coordinates. We determine the geodesic equations on the extended Siegel--Jacobi upper half-plane $\tilde{\mathcal{X}}^J_1 =\frac{G^J_1(\R)}{\rm{SO}(2)}\approx\mathcal{X}^J_1\times\mathbb{R}\approx \mathcal{X}_1 \times\mathbb{R}^3$, where $\mathcal{X}^J_1$ ($\mathcal{X}_1)$ denotes the Siegel-Jacobi upper half-plane (respectively Siegel upper half-plane). Equating successively with…
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 states give the same equations of quantum and classical motion when are expressed in the coordinates in which the Kähler two-form $ω_{\mathcal{D}^J_1} $ can be written as $ω_{\mathcal{D}^J_1}=ω_{\mathcal{D}_1}+ω_{\mathbb{C}^1}$. The Wei-Norman equations on $\mathcal{D}^J_1$ are a…
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 orthonormal bases of the representation spaces are given.
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.
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 operators. We present problems which are natural. We give some new partial solutions for these natural problems.
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 the Siegel-Jacobi disk $\mathcal{D}^J_1$ are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel-Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding, and th…
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 $\mathcal{D}^J_1$ determined by a linear Hamiltonian in the generators of the Jacobi group $G^J_1=H_1\rtimes\text{SU}(1,1)$ is described by a Riccati equation on $\mathcal{D}_1$ and a linear first order differential equation in $z\in\mathbb{C}$, where $H_1$ denotes the real 3-dimensional Heisenbe…
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 hermitian linear Hamiltonian in the generators of the Jacobi group $G^J_n=H_n\rtimes\text{Sp}(n,\R)_{\C}$ are described by a matrix Riccati equation on $\mc{D}_n$ and a linear first order differential equation in $z\in\C^n$, with coefficients depending also on $W\in\mc{D}_n$. $H_n$ deno…
The real Jacobi group $G^J_n(\mathbb{R})$, defined as the semidirect product of the Heisenberg group ${\rm H}_n(\R)$ with the symplectic group ${\mr {Sp}}(n,\mathbb{R})$, admits a matrix embedding in $\text{Sp}(n+1,\mathbb{R})$. The modified pre-Iwasawa decomposition of $\rm{Sp}(n,\mathbb{R})$ allows us to introduce a convenient coordinatization $S_n$ of $G^J_n(\mathbb{R})$, which for $G^J_1(\mathbb{R})$ coincides with the $S$-coordinates. Invariant one-forms on $G^J_n(\mathbb{R})$ are determined. The formula of the 4-parameter invariant metric on $G^J_1(\R)$ obtained as sum of squares of 6 in…
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-invariant vector fields are calculated in the S-coordinates $(x,y,θ,p,q,κ)$, and a left-invariant metric depending of 4 parameters $(α,β,γ,δ)$ is obtained. An invariant metric depending of $(α,β)$ in the variables $(x,y,θ)$ on the Sasaki manifold ${\rm SL}(2,\mathbb{R})$ is presen…
We determine the Hamiltonian vector field on an odd dimensional manifold endowed with almost cosymplectic structure. This is a generalization of the corresponding Hamiltonian vector field on manifolds with almost transitive contact structures, which extends the contact Hamiltonian systems. Applications are presented to the equations of motion on a particular five-dimensional manifold, the extended Siegel-Jacobi upper-half plane $\tilde{\mathcal{X}}^J_1$. The $\tilde{\mathcal{X}}^J_1$ manifold is endowed with a generalized transitive almost cosymplectic structure, an almost cosymplectic structu…
It is shown that Nambu-Poisson and Nambu-Jacobi brackets can be defined inductively: a n-bracket, n>2, is Nambu-Poisson (resp. Nambu-Jacobi) if and only if fixing an argument we get a (n-1)-Nambu-Poisson (resp. Nambu-Jacobi) bracket. As a by-product we get relatively simple proofs of Darboux-type theorems for these structures.
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontriv…
