论文标题
关于扩展的Siegel的几何形状 - Jacobi上半平面
Remarks on the geometry of the extended Siegel--Jacobi upper half-plane
论文作者
论文摘要
真正的jacobi组$ g^j_1（\ mathbb {r}）= {\ rm sl}（2，2，\ mathbb {r}）\ ltimes {\ rm h} _1 $，其中$ {\ rm h}，其中3二维_ $ yes $ sagot in $ sogiation $ samearized ins pogair sagy sagy sagy sagy sagy sagy $（x，y，θ，p，q，κ）$。我们表明，基于siegel-jacobi disk $ \ mathcal {d}^j_1 $从perelomov的非均衡相干状态向量出现的参数$η$。 siegel上的两参数不变式指标 - $（x，y，\ rm {re} 〜η，\ rm {im} 〜η）$。事实证明，五维流形$ \ tilde {\ Mathcal {x}}^j_1 = \ frac {g^j_1（\ r）} {\ rm {so}（so}（so}（2）}}}}}半平面是相对于$ g^j_1（\ mathbb {r}）$的三参数度量不变性的还原性，非对称性，非自然还原的多种流形的，并且确定了其测量矢量。
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.
