Long Guang HE
Let (Γ P, α, β) be a Poisson groupoid. If its each α-fiber intersects each βfiber not more than one point, then for any x ∈Γ, α(x) = u, β(x) = v, the characteristic distribution of Γ has a decomposition into direct product △(x)= △α(x) + △β(x),such that △α(x) tangents to α-fiber, △β(x) tangents to β-fiber, △α(x) and △β(x) are both Symplectic subspaces of △(x). Since the complete integrabilities of △α(x) and △β(x), there exist Symplectic submanifolds S and S of Symplectic leaf S, such that α: S→α(S) Su is Symplectic diffeomorphism and β: S→β(S) Sv is anti-Symplectic diffeomorphism. Where Su and Sv are Symplectic leaves of P.For general Poisson groupoids, we can obtain S and S with similar properties.Using these results to Symplectic groupoids, we can obtain some more concrete results.