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A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for are smoothly equivalent provided their union is again a smooth atlas for .
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 ...
Introduction to smooth manifolds. New York: Springer. ISBN 0-387-95448-1. A textbook on manifold theory. See also the same author's textbooks on topological manifolds (a lower level of structure) and Riemannian geometry (a higher level of structure).
Conversely, given any contact manifold M, the product M×R has a natural structure of a symplectic manifold. If α is a contact form on M, then ω = d(e t α) is a symplectic form on M×R, where t denotes the variable in the R-direction. This new manifold is called the symplectization (sometimes symplectification in the literature) of the ...
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.
The jet space can be endowed with a smooth structure (i.e. a structure as a C ∞ manifold) which make it into a topological space. This topology is used to define a topology on C ∞ ( M , N ). For a fixed integer k ≥ 0 consider an open subset U ⊂ J k ( M , N ), and denote by S k ( U ) the following:
Given a Lie group action of on , the orbit space / does not admit in general a manifold structure. However, if the action is free and proper, then M / G {\displaystyle M/G} has a unique smooth structure such that the projection M → M / G {\displaystyle M\to M/G} is a submersion (in fact, M → M / G {\displaystyle M\to M/G} is a principal G ...
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.