Darboux's theorem
/ dar-BOO /
Riemannian geometry has curvature: even in the smallest patch, a curved surface is not flat, and curvature is a local fingerprint that distinguishes one geometry from another. Darboux's theorem says symplectic geometry is dramatically different. Zoom in anywhere on any symplectic manifold and it looks exactly like the standard model R^{2n} with omega = sum dp_i ^ dq_i. There is no local fingerprint at all — every symplectic manifold of a given dimension is locally indistinguishable from every other.
Precisely, Darboux's theorem states: if (M, omega) is a symplectic manifold of dimension 2n, then around every point p there exist local coordinates (q_1, ..., q_n, p_1, ..., p_n), called Darboux coordinates, in which omega = sum_{i=1}^n dp_i ^ dq_i. The proof is a slick application of the Moser trick: interpolate omega_t = omega_0 + t(omega_1 - omega_0) between the actual form and the model, and use the closedness of both to solve for a time-dependent vector field whose flow pulls one onto the other. Closedness (d omega = 0) is exactly the hypothesis that makes this PDE solvable; nondegeneracy lets you invert omega to define that vector field.
The consequence reshapes the whole subject. Because there are no local invariants — no symplectic analogue of curvature — all of symplectic geometry's content is global and topological. Questions like 'can this region be embedded in that one?' (Gromov nonsqueezing) or 'how many fixed points must this map have?' (Arnold conjecture, Floer theory) replace the local curvature questions of Riemannian geometry. A frequent misconception is to look for 'local symplectic curvature' or a pointwise symplectic invariant: there is none, and Darboux's theorem is precisely the statement that the search is futile.
Take any symplectic surface (a 2-manifold with an area form omega). Darboux says near any point you can find coordinates (q, p) with omega = dp ^ dq — i.e. the area form becomes the standard one. Concretely, on the sphere S^2 with its round area form, no point is 'specially curved' symplectically: every point has a neighborhood symplectomorphic to a flat patch of (R^2, dp ^ dq). Contrast this with the round metric on S^2, where Gaussian curvature is a nonzero pointwise invariant everywhere.
Every symplectic patch is the standard model: omega = sum dp_i ^ dq_i locally, no curvature to detect.
Darboux is purely local. It does NOT say two symplectic manifolds are globally symplectomorphic, nor that a symplectic embedding exists between given regions — those global questions are precisely where nonsqueezing and capacities have content. Local triviality coexists with rich global rigidity.