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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 .
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, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.
Important to applications in mathematics and physics [1] is the notion of a flow on a manifold. In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to .
It follows that is a regular value of , so () and its quotient / are both smooth manifolds. The quotient inherits a symplectic form from M {\displaystyle M} ; that is, there is a unique symplectic form on the quotient whose pullback to μ − 1 ( 0 ) {\displaystyle \mu ^{-1}(0)} equals the restriction of ω {\displaystyle \omega } to μ − 1 ...
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).
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:
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.