, together with the canonical symplectic form of . A Lie-Poisson structure on is integrable if and only if the Lie algebroid is integrable to a Lie groupoid , the symplectic groupoid being the cotangent groupoid with the canonical symplectic form. Submanifolds A Poisson submanifold of is an immersed submanifold such that the immersion map is a Poisson map. Equivalently, one asks that every Hamiltonian vector field , for , is tangent to . This definition is very natural and satisfies several good properties, e.g. the transverse intersection of two Poisson submanifolds is again a Poisson submanifold. However, it has also a few problems: Poisson submanifolds are rare: for instance, the only Poisson submanifolds of a symplectic manifold are the open sets; the definition does not behave functorially: if is a Poisson map transverse to a Poisson submanifold of , the submanifold of is not necessarily Poisson. In order to overcome these problems, one often uses the notion of a Poisson transversal (originally called cosymplectic submanifold). This can be defined as a submanifold which is transverse to every symplectic leaf and such that the intersection is a symplectic submanifold of . It follows that any Poisson transversal inherits a canonical Poisson structure from . In the case of a nondegenerate Poisson manifold (whose only symplectic leaf is itself), Poisson transversals are the same thing as symplectic submanifolds. More general classes of submanifolds play an important role in Poisson geometry, including Lie-Dirac submanifolds, Poisson-Dirac submanifolds, coisotropic submanifolds and pre-Poisson submanifolds. See also Nambu-Poisson manifold Poisson–Lie group Poisson supermanifold Kontsevich quantization formula References Books and surveys Previous version available on . See also the review by Ping Xu in the Bulletin of the AMS. Differential geometry Symplectic geometry Smooth manifolds Structures on manifolds