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  2. 4-manifold - Wikipedia

    en.wikipedia.org/wiki/4-manifold

    Classify the different smooth structures on a smoothable manifold. There is an almost complete answer to the first problem asking which simply connected compact 4-manifolds have smooth structures. First, the Kirby–Siebenmann class must vanish. If the intersection form is definite Donaldson's theorem (Donaldson 1983) gives a complete answer ...

  3. Seiberg–Witten invariants - Wikipedia

    en.wikipedia.org/wiki/Seiberg–Witten_invariants

    A manifold M is said to be of simple type if the Seiberg–Witten invariant vanishes whenever the expected dimension of the moduli space is nonzero. The simple type conjecture states that if M is simply connected and b 2 + (M) ≥ 2 then the manifold is of simple type. This is true for symplectic manifolds.

  4. Ricci flow - Wikipedia

    en.wikipedia.org/wiki/Ricci_flow

    Given a smooth manifold M and an open real interval (a, b), a Ricci flow assigns, to each t in the interval (a,b), a Riemannian metric g t on M such that ⁠ ∂ / ∂t ⁠ g t = −2 Ric g t. The Ricci tensor is often thought of as an average value of the sectional curvatures, or as an algebraic trace of the Riemann curvature tensor. However ...

  5. Hodge theory - Wikipedia

    en.wikipedia.org/wiki/Hodge_theory

    In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.

  6. Generalized Stokes theorem - Wikipedia

    en.wikipedia.org/wiki/Generalized_Stokes_theorem

    Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.

  7. Lie group action - Wikipedia

    en.wikipedia.org/wiki/Lie_group_action

    However, if the action is free and proper, then / has a unique smooth structure such that the projection / is a submersion (in fact, / is a principal -bundle). [ 2 ] The fact that M / G {\displaystyle M/G} is Hausdorff depends only on the properness of the action (as discussed above); the rest of the claim requires freeness and is a consequence ...

  8. Real projective space - Wikipedia

    en.wikipedia.org/wiki/Real_projective_space

    Real projective spaces are smooth manifolds. On S n, in homogeneous coordinates, (x 1, ..., x n+1), consider the subset U i with x i ≠ 0. Each U i is homeomorphic to the disjoint union of two open unit balls in R n that map to the same subset of RP n and the coordinate transition functions are smooth. This gives RP n a smooth structure.

  9. Symplectic manifold - Wikipedia

    en.wikipedia.org/wiki/Symplectic_manifold

    Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...