By S. T Hu
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Although we know from the inverse function theorem that the map S is one-to-one on some neighborhood of the zero section of the bundle G X G, n°, we don't know whether this neighborhood can be chosen to be G-invariant. On the other hand there exists a G-invariant neighborhood of the zero section consisting of the points at which the Jacobian of E is bijective. Indeed, since the map E is G-equivariant and since it is an embedding on each fiber, a point [g, v] is a regular point of the map if and only if the orbit through the image S([g, v]) is transverse to the image of the fiber g (A + n°).
4 (base change) Let,-r: X -+ B be a symplectic fibration, let B1 be a manifold, and let f: BI B be a smooth map. 10) B If r is a connection on X we can pull it back to get a connection r, on X, which will be symplectic if r is. Furthermore the forms pull back consistently to give wr, = g*wr. 4 let us take B1 = T*B. Let wB be the standard symplectic form on T* B. 9) to obtain the form wr, + rri WB. 12) This is clearly closed and r l -compatible. 12) is always symplectic. 12) is used in physics to adjoin "internal variables" to a classical dynamical system.
In other words, in the weak coupling limit there is a unique symplectic structure on M compatible with the symplectic structure on fiber and base. 2 Examples of Symplectic Fibrations: The Coadjoint Orbit Hierarchy As we pointed out in the introduction the purpose of this monograph is to explore connections between two subjects which seem to have, on the face of it, little to do with each other: symplectic fibrations and the multiplicity diagrams associated with representations of Lie groups. What will supply the bridge between these two topics is the coadjoint orbit hierarchy.