# 泊松流形

（重定向自泊松双向量

## 定义

M 上一个泊松结构Poisson structure）是一个双线性映射

${\displaystyle \{,\}:C^{\infty }(M)\times C^{\infty }(M)\to C^{\infty }(M),\,}$

${\displaystyle \{f,g\}=-\{g,f\},\,}$

${\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0,\,}$

C(M) 关于第一个变量的导子

${\displaystyle \{fg,h\}=f\{g,h\}+g\{f,h\}}$  对所有 ${\displaystyle f,g,h\in C^{\infty }(M).\,}$

${\displaystyle X_{g}(f)=\{f,g\}\,}$

${\displaystyle B_{M}:\mathrm {T} ^{*}M\to \mathrm {T} M,\,}$

## 泊松双向量

${\displaystyle \{f,g\}=\langle \mathrm {d} f\otimes \mathrm {d} g,\eta \rangle ,\,}$

${\displaystyle \eta _{x}=\sum _{i,j=1}^{m}\eta _{ij}(x){\frac {\partial }{\partial x_{i}}}\otimes {\frac {\partial }{\partial x_{j}}}\,}$

${\displaystyle \{f,g\}(x)=\sum _{i,j=1}^{m}\eta _{ij}(x){\frac {\partial f}{\partial x_{i}}}\otimes {\frac {\partial g}{\partial x_{j}}}.\,}$

## 泊松映射

${\displaystyle \{f_{1},f_{2}\}_{N}\circ \phi =\{f_{1}\circ \phi ,f_{2}\circ \phi \}_{M}\,}$

## 乘积流形

${\displaystyle \{f_{1},f_{2}\}_{M\times N}(x,y)=\{f_{1}(x,\cdot ),f_{2}(x,\cdot )\}_{N}(y)+\{f_{1}(\cdot ,y),f_{2}(\cdot ,y)\}_{M}(x)\,}$

${\displaystyle f(\cdot ,\cdot ):M\times N\to \mathbb {R} ,\,}$

${\displaystyle f(x,\cdot ):N\to \mathbb {R} \,}$

${\displaystyle f(\cdot ,y):M\to \mathbb {R} .\,}$

## 例子

${\displaystyle \{f_{1},f_{2}\}(x)=\langle \;\left[(df_{1})_{x},(df_{2})_{x}\right]\,,x\rangle }$

${\displaystyle \eta _{ij}(x)=\sum _{k}c_{ij}^{k}\langle x,e_{k}\rangle \,}$

## 复结构

${\displaystyle \left(J\otimes J\right)(\eta )=\eta .\,}$

## 参考文献

• A. Lichnerowicz, "Les variétès de Poisson et leurs algèbres de Lie associées", J. Diff. Geom. 12 (1977), 253-300.
• A. A. Kirillov, "Local Lie algebras", Russ. Math. Surv. 31 (1976), 55-75.
• V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambridge Univ. Press 1984.
• P. Liberman, C.-M. Marle, Symplectic geometry and analytical mechanics, Reidel 1987.
• K. H. Bhaskara, K. Viswanath, Poisson algebras and Poisson manifolds, Longman 1988, ISBN 0-582-01989-3.
• I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994. See also the review by Ping Xu in the Bulletin of the AMS.