Abstract
Let M be a manifold. Denote the degree p skew symmetric contravariant differentiable tensor field space on M by \(D_p(M)\).
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- 1.
Added by the authors of the Forewords. Even when the distribution L is not of constant rank, it is completely integrable in a generalized sense. Its maximal integral manifolds, called here the leaves of the Poisson manifold (M, w), are called, in other texts, the symplectic leaves of that Poisson manifold. The leaves of (M, w) are in general immersed, not embedded, submanifolds of M. The proof that for each point \(x\in M\) there exists a unique symplectic leaf which contains that point can be done either by working with local quotients of that Poisson manifold, or by application of a generalization of Frobenius’ theorem proven around 1973 independently by P. Stefan (Integrability of systems of vectorfields, J. London Math. Soc., 2–21(3), pp. 544–556, 1980) and H. Sussmann (Orbits of families of vector fields and integrability of systems with singularities, Bull. Amer. Math. Soc., 79(1):197–199, 1973).
- 2.
Added by the authors of the Forewords. This Poisson structure which exists on the dual space of a Lie algebra was noticed by Sophus Lie and rediscovered, much later, independently by A. Kirillov, B. Kostant and J.-M. Souriau. It is often called the canonical Lie-Poisson structure, or the Kirillov-Kostant-Souriau structure.
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Koszul, JL., Zou, Y.M. (2019). Poisson Manifolds. In: Introduction to Symplectic Geometry. Springer, Singapore. https://doi.org/10.1007/978-981-13-3987-5_5
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DOI: https://doi.org/10.1007/978-981-13-3987-5_5
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