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Download QR code; Print/export Download as PDF; Printable version; ... For more examples see 4-manifold. Special types of manifolds. Manifolds related to spheres
Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:
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.
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.
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non- simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups . [ 1 ]
There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants. For instance, for orientable surfaces, the classification of surfaces enumerates them as the connected sum of tori, and an invariant that classifies them is the genus or Euler characteristic.
Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. [1] However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve ...