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 $O(n)$

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 as differentiable manifolds and Riemannian manifolds.

Mathematical Definition

Topological Manifold

In topology, a manifold of dimension $n$, or an n-manifold, is defined as a Hausdorff space where an open neighbourhood of each point is homeomorphic (i.e. there exists a smooth bijective map from the manifold with a smooth inverse) 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 $\left(U_{\alpha },\psi _{\alpha }\right)$ for some $\scriptstyle \alpha$ varying over an index set $A$ such that

1. $U_{\alpha }\in M,\quad M=\bigcup _{\alpha \in A}U_{\alpha }$
2. $\psi _{\alpha }$ maps $U_{\alpha }$ bijectively to an open set $\scriptstyle V_{\alpha }\in \,\mathbb {R} ^{n}$, and for $\scriptstyle \alpha ,\,\beta \,\in A$ the image $\scriptstyle \psi (U_{\alpha }\cap U_{\beta })\,\in \,\mathbb {R} ^{n}$ is an open set. The function $\psi _{\alpha }:U_{\alpha }\rightarrow V_{\alpha }$ is called a chart.
3. For $\scriptstyle \alpha ,\,\beta \,\in A$, the coordinate change is a differentiable map between two open sets in $\scriptstyle \mathbb {R} ^{n}$ whereby $\qquad \psi _{\beta }\circ \psi _{\alpha }^{-1}:\psi _{\alpha }\left(U_{\alpha }\cap U_{\beta }\right)\rightarrow \psi _{\beta }(U_{\alpha }\cap U_{\beta }).$

The set M is a differentiable manifold if and only if it comes equipped with a countable atlas and satisfies the Hausdorff property. The important difference between a differentiable manifold and a topological manifold is that the charts are diffeomorphisms (a differentiable function with a differentiable inverse) rather than homeomorphisms.

Differentiable manifolds have a tangent space $T_{p}M$, the space of all tangent vectors, associated with each point $p$ on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within $\scriptstyle \mathbb {R} ^{n+1}$, it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible.

Riemannian Manifolds

To define distances and angles on a differentiable manifold, it is necessary to define a metric. A differentiable manifold equipped with a metric is called a Riemannian manifold. A Riemannian metric is a generalisation of the usual idea of the scalar or dot product to a manifold. In other words, a Riemannian metric $g=\{g_{p}\}_{p\in M}$ is a set of symmetric inner products

$g_{p}:T_{p}M\times T_{p}M\rightarrow \mathbb {R}$

which depend smoothly on $p$.

See Also

• Tangent vectors
• Differential Geometry
• Riemannian Geometry