Manifold

topology analysis geometry

Definition

Manifold

An $n$-dimensional manifold is a topological space that locally resembles $n$-dimensional Euclidean space ${\mathbb{R}}^n$ at each point.

Formally, a space $M$ is an $n$-manifold if for every point $p\in M$, there exists an open neighbourhood $U$ containing $p$ that is homeomorphic to an open subset of ${\mathbb{R}}^n$. The homeomorphism $\varphi :U\to {\mathbb{R}}^n$ is called a coordinate chart.

Structure

Charts and Atlases: A collection of charts $\{{\varphi }_i:U_i\to {\mathbb{R}}^n\}$ that covers the entire manifold is called an atlas.

Transition Maps: On the overlaps of charts, the maps ${\varphi }_j\circ {\varphi }_i^{-1}$ are mappings between subsets of ${\mathbb{R}}^n$. If these transition maps are smooth ($C^{\infty }$), $M$ is called a differentiable manifold.

Examples

The $n$-Sphere ($S^n$): The surface of a ball in ${\mathbb{R}}^{n+1}$.

The $n$-Torus ($T^n$): A generalisation of the surface of a doughnut.

Projective Spaces: Spaces of lines through the origin in ${\mathbb{R}}^{n+1}$.

Euclidean Space: ${\mathbb{R}}^n$ is itself an $n$-manifold with a single chart (the identity map).