Def
A manifold is a space that:
- Locally looks like flat Euclidean space (like a line, plane, or 3D space),
- But globally may have curvature or a more complex shape.
Think of a manifold as a generalisation of lines, planes, and 3D space that allows bending or twisting, as long as every small patch still looks flat. A small patch of Earth looks like a plane, but the whole Earth is approximately spherical.
Formally, an -dimensional manifold is a topological space where each point has a neighbourhood homeomorphic to . This lets us use local coordinates, called charts, even when one global coordinate system is awkward or impossible.
Metric and geometry
A bare manifold tells us about neighbourhoods and continuity, but not lengths or angles. A metric adds measurement. Once a metric is present, we can define distances, volumes, geodesics, and Curvature.
Examples
| Object | Type | Local geometry | Global geometry |
|---|---|---|---|
| Line | 1D manifold | flat, infinite | |
| Circle | 1D manifold | locally line-like | loops back on itself |
| Plane | 2D manifold | flat | |
| Sphere | 2D manifold | locally planar | positively curved |
| Spacetime | 4D manifold | locally Minkowski-like | curved by energy/momentum |
Why it matters
Manifolds are the bridge from ordinary Euclidean spaces to modern physics. In general relativity, spacetime is a 4D manifold whose metric curvature describes gravity; see Gravity as emergent from spacetime curvature and Cosmology index. Calculus on manifolds generalises Vector Calculus index.