Manifolds

Definition

A \(n\)-dimensional manifold \(\mathcal{M}\) is a space that can locally be approximated by \(\mathbb{R}^n\), the Euclidean space of dimension \(n\). We can think of a manifold as a surface that may be curved or have a complex structure, but at small enough scale, it resembles flat Euclidean space.

An example of this is the Earth’s surface, which is a 2-dimensional manifold. Locally, it can be approximated by flat planes (like a map), but globally it is curved (like a sphere).