Manifold

Differentiable manifolds

It is easy to define the notion of a topological manifold, but it is very hard to work with this object. The smooth manifold defined below works better for most applications, in particular it makes possible to apply "calculus" on the manifold.

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We start with a topological manifold M without boundary. An open set of M together with a homeomorphism between the open set and an open set of Rn is called a coordinate chart.

Related Topics:
Open set - Coordinate chart

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A collection of charts which cover M is called an atlas of M.

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The homeomorphisms of two overlapping charts provide a transition map from a subset of Rn to some other subset of Rn.

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If all these maps are k times continuously differentiable, then the atlas is an Ck atlas.

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Example: The unit sphere in R3 can be covered by two charts: the complements of the north and south poles with coordinate maps - stereographic projections relative to the two poles.

Related Topics:
Sphere - Stereographic projection

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Atlas

Two Ck atlases are called equivalent if their union is a Ck atlas.

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This is an equivalence relation, and a Ck manifold is defined to be a manifold together with an equivalence class of Ck atlases. If all the connecting maps are infinitely often differentiable, then one speaks of a smooth or

Related Topics:
Equivalence relation - Equivalence class

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C∞ manifold; if they are all analytic, then the manifold is an analytic or Cω manifold.

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Intuitively, a smooth atlas provides local coordinate systems such that the change-of-coordinate functions are smooth.

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These coordinate systems allow one to define differentiability and integrability of functions on M.

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Once a C1 atlas on a paracompact manifold is given, we can refine it to a real analytic atlas (meaning that the new atlas, considered as a C1 atlas, is equivalent to the given one), and all such refinements give the same analytic manifold. Therefore, one often considers only these latter manifolds.

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Not every topological manifold admits such a smooth atlas. The lowest dimension is 4 where there are non-smoothable topological manifolds. Also, it is possible for two non-equivalent differentiable manifolds to be homeomorphic. The famous example was given by John Milnor of exotic 7-spheres, i.e. non-diffeomorphic topological 7-spheres.

Related Topics:
Homeomorphic - John Milnor - Exotic 7-spheres

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The Hausdorff assumption

Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.

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Homogenous, second-countable and paracompact

A manifold is said to be homogeneous for its homeomorphism group, or diffeomorphism group, if that group acts transitively on it; this is true for connected manifolds. Thus every connected manifold without boundary is homogeneous.

Related Topics:
Homogeneous - Diffeomorphism

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It can be shown that a manifold is metrizable if and only if it is paracompact. Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an n-manifold.

Related Topics:
Metrizable - Paracompact - Long line - Pathological

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Sometimes n-manifolds are defined to be second-countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.

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Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.

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