Manifolds Explained

Manifolds are topological spaces that locally resemble Euclidean space, providing a framework for studying higher dimensions.

Definition

An $n$-manifold is a space where every point has a neighborhood homeomorphic to ${\mathbb{R}}^n$, equipped with a topology.

Examples

The sphere $S^2$ (2-manifold) looks like ${\mathbb{R}}^2$ locally; the torus is another 2-manifold.

Applications

Manifolds model spacetime in general relativity and phase spaces in dynamical systems.