Manifold

topology analysis geometry

Definition

Manifold

An -dimensional manifold is a topological space that locally resembles -dimensional Euclidean space at each point.

Formally, a space is an -manifold if for every point , there exists an open neighbourhood containing that is homeomorphic to an open subset of . The homeomorphism is called a coordinate chart.

Structure

Charts and Atlases: A collection of charts that covers the entire manifold is called an atlas.

Transition Maps: On the overlaps of charts, the maps are mappings between subsets of . If these transition maps are smooth (), is called a differentiable manifold.

Examples

The -Sphere (): The surface of a ball in .

The -Torus (): A generalisation of the surface of a doughnut.

Projective Spaces: Spaces of lines through the origin in .

Euclidean Space: is itself an -manifold with a single chart (the identity map).