Let L and L’ be two Lagrangians in a symplectic manifold M, intersecting transversely at some point x. It is often stated that, in a Darboux chart about x, one can arrange for L and L’ to be the standard R^n and i R^n inside of C^n. This means:
- Tangentially, the symplectic group acts transitively on pairs of transversely intersecting linear Lagrangian subspaces.
- Locally, one can do even better, deforming the Lagrangians in M to look “straight.”
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