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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 ...
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.
In this case, (,) is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.
Stochastic differential geometry provides insight into classical analytic problems, and offers new approaches to prove results by means of probability. For example, one can apply Brownian motion to the Dirichlet problem at infinity for Cartan-Hadamard manifolds [4] or give a probabilistic proof of the Atiyah-Singer index theorem. [5]
This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa. In general, computations with the maximal atlas of a manifold are rather unwieldy.
Recall that a topological manifold is a topological space which is locally homeomorphic to . Differentiable manifolds (also called smooth manifolds) generalize the notion of smoothness on in the following sense: a differentiable manifold is a topological manifold with a differentiable atlas, i.e. a collection of maps from open subsets of to the manifold which are used to "pull back" the ...
The objects of Man • p are pairs (,), where is a manifold along with a basepoint , and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. : (,) (,), such that () =. [1] The category of pointed manifolds is an example of a comma category - Man • p is exactly ({}), where {} represents an arbitrary singleton ...
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.