Instituto Superior Técnico Symplectic Geometry Luís Ferreira IST no. 31727 · 2 / 2 / 93 Abstract We examine what survives of symplectic geometry when the non-degeneracy of \(\omega\) is given up, and how far the algebra of observables can still be built. Non-degeneracy is what makes \(\omega^\flat:TM\to T^*M\) an isomorphism, and therefore what assigns to each observable \(g\) a single field \(X_g\) with \(\iota_{X_g}\omega=dg\); once it fails, \(X_g\) is determined only up to the kernel of \(\omega^\flat\) and the Poisson bracket loses its meaning. We show that if \(\omega\) has constant rank the kernel is a subbundle of \(TM\) which, because \(\omega\) is closed, is involutive and hence arises from a regular foliation . Taking the quotient of \(TM\) by the kernel directions, and of \(M\) by the leaves , yields a base which is again a symplectic manifold , and on it the Poisson brackets, the canonical transformations and the remaining structure are ...
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