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.