Every (paracompact Hausdorff) differentiable manifold can be equipped with a Riemannian structure. Therefore the existence of a metric does not impose constraints on a topology 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.
Last revised on June 2, 2024 at 23:50:29. See the history of this page for a list of all contributions to it.