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 Rn, equipped with a topology.

Examples

The sphere S2 (2-manifold) looks like R2 locally; the torus is another 2-manifold.

Applications

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