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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]
If M is a piecewise linear manifold then the problem of finding the compatible smooth structures on M depends on knowledge of the groups Γ k = Θ k. More precisely, the obstructions to the existence of any smooth structure lie in the groups H k+1 ( M , Γ k ) for various values of k , while if such a smooth structure exists then all such ...
In mathematics, differential topology is the field dealing with the topological properties and smooth properties [a] of smooth manifolds.In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
Download as PDF; Printable version; In other projects ... Pages in category "Smooth manifolds" ... Almost complex manifold;
Every smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds , which (in four dimensions) are the main objects of the theory of general relativity .
For smooth manifolds M the problem reduces to finding the form of the homomorphism () (), where () is the oriented bordism group of X. [4] The connection between the bordism groups and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms ( ()) ().
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.
Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists of a collection {} of vector subspaces with the following property: Around any there exist a neighbourhood and a collection of vector fields, …, such that, for any point , span {(), …, ()} =.