Poisson嵌入子流形的一个性质

A Property on Poisson Embedding Submanifolds

讨论了Ｐｏｉｓｏｎ子流形的一个有趣的性质：若Ｐ是Ｐｏｉｓｏｎ流形（Ｑ，ＷＱ）的Ｐｏｉｓｏｎ嵌入子流形，则在每一点ｘ∈Ｐ处，通过适当选择可使Ｐ的横截Ｐｏｉｓｏｎ流形Ｎ１恰是Ｑ的横截Ｐｏｉｓｏｎ流形的Ｐｏｉｓｏｎ嵌入子流形．反之，若Ｐ是（Ｑ，ＷＱ）的嵌入子流形，且在每一点ｘ∈Ｐ处，存在ｘ在Ｑ中的邻域Ｕ和直积分解Ｕ＝Ｓ×Ｎ１×Ｎ２，使得Ｓ是辛叶，Ｎ１×Ｎ２是横截Ｐｏｉｓｏｎ流形，Ｓ×Ｎ１是Ｐ中ｘ的邻域，Ｎ１是Ｎ１×Ｎ２的Ｐｏｉｓｏｎ子流形。

The paper discusses an interesting property of Poisson submanifolds: If P isa Poisson embedding submanifold in (Q,W Q ), then at every point x∈P ,the transversal Poisson manifold N 1 in P is just Poisson embedding submanifold in the transversal Poisson manifold N in Q . Inversely, if P is a embedding submanifold in ( Q,W Q ), and at every point x∈P, there is a neighborhood U in Q and decomposition into direct product U=S×N 1×N 2 ,such that S is sympplectic leaf, N 1×N 2 is the transversal Poisson manifold, S×N 1 is the neighborhood at x in P,N 1 is Poisson submanifold in N 1×N 2, then P is Poisson submanifold in Q .