Manifold (geometry)

A manifold is an abstract mathematical space that looks locally like Euclidean space, but globally may have a very different structure. An example of this is a sphere: if one is very close to the surface of the sphere, it looks like a flat plane, but globally the sphere and plane are very different. Other examples of manifolds include lines and circles, and more abstract spaces such as the orthogonal group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(n)}

The concept of a manifold is very important within mathematics and physics, and is fundamental to certain fields such as differential geometry, Riemannian geometry and General Relativity.

The most basic manifold is a topological manifold, but additional structures can be defined on the manifold to create objects such differentiable manifolds and Riemannian manifolds.

Mathematical Definition

Topological Manifold

In topology, a manifold of dimension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , or an n-manifold, is defined as a Hausdorff space where an open neighbourhood of each point is homeomorphic to $\scriptstyle \mathbb {R} ^{n}$.

Differentiable Manifold

To define differentiable manifolds, the concept of an atlas, chart and a coordinate change need to be introduced. An atlas of the Earth uses these concepts: the atlas is a collection of different overlapping patches of small parts of a spherical object onto a plane. The way in which these different patches overlap is defined by the coordinate change.

Let M be a set. An atlas of M is a collection of pairs $\scriptstyle \left(U_{\alpha },\psi _{\alpha }\right)$ for some $\scriptstyle \alpha$ varying over an index set $A$ such that

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle U_{\alpha} \in M, \quad M = \bigcup_{\alpha \in A} U_{\alpha} }

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{\alpha} } maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{\alpha}} bijectively to an open set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle V_{\alpha} \in \, \mathbb{R}^n} , and for $\scriptstyle \alpha ,\,\beta \,\in A$ the image Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \psi(U_{\alpha} \cap U_{\beta}) \, \in \, \mathbb{R}^n} is an open set. The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \psi_{\alpha}: U_{\alpha} \rightarrow V_{\alpha} } is called a chart.

• For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha, \, \beta \, \in A} , the coordinate change is a differentiable map between two open sets in $\scriptstyle \mathbb {R} ^{n}$ whereby

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{\beta} \circ \psi_{\alpha}^{-1}: \psi_{\alpha} \left(U_{\alpha} \cap U_{\beta} \right) \rightarrow \psi_{\beta} (U_{\alpha} \cap U_{\beta})}

Then the set M is a differentiable manifold if and only if it comes equipped with a countable atlas and satisfies the Hausdorff property.