Search results
Results from the WOW.Com Content Network
These homeomorphisms are the charts of the manifold. A topological manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any particular ...
It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact [3] or second-countable. [2] In the remainder of this article a manifold will mean a topological manifold. An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to R n.
This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics . For categorical listings see Category:Manifolds and its subcategories.
In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a (G, X)-structure are always manifolds and are called (G, X)-manifolds.
An atlas for a topological space is an indexed family {(,):} of charts on which covers (that is, =).If for some fixed n, the image of each chart is an open subset of n-dimensional Euclidean space, then is said to be an n-dimensional manifold.
The study of maps of 1-dimensional manifolds are a non-trivial area. For example: Groups of diffeomorphisms of 1-manifolds are quite difficult to understand finely [2] Maps from the circle into the 3-sphere (or more generally any 3-dimensional manifold) are studied as part of knot theory.
There are three main types of structures important on manifolds. The foundational geometric structures are piecewise linear, mostly studied in geometric topology, and smooth manifold structures on a given topological manifold, which are the concern of differential topology as far as classification goes. Building on a smooth structure, there are:
Manifolds in contemporary mathematics come in a number of types. These include: smooth manifolds, which are basic in calculus in several variables, mathematical analysis and differential geometry; piecewise-linear manifolds; topological manifolds. There are also related classes, such as homology manifolds and orbifolds, that resemble manifolds.