Every (paracompact Hausdorff) differentiable manifold can be equipped with a Riemannian structure. Therefore the existence of a metric does not impose constraints on a toplogy of a manifold. The local Riemannian structure of a Riemannian manifold on the other hand can be very different with the same global topology: there are even locally many nonisometric Riemannian structures on the same manifold/neighborhood. All even dimensional manifolds allow *locally* a (unique up to isomorphism) symplectic structure. Symplectic manifolds, due Darboux theorem, are all locally isomorphic, but according to the spectacular findings of Gromov in 1985, there are still global invariants of symplectic structure as well as topological constraints on even-dimensional manifolds permitting symplectic structure.

- Dusa McDuff, Dietmar Salamon,
*Introduction to symplectic topology*, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp. - Yong-Geun Oh,
*Symplectic topology and Floer homology*, pdf

*Pseudo holomorphic curves in symplectic manifolds*, Inventiones Mathematicae**82**, n. 2 (1985) 307–347.

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