Manifolds
=========

Definition
----------

A :math:`n`-dimensional manifold :math:`\mathcal{M}` is a space that can locally be approximated by :math:`\mathbb{R}^n`, the Euclidean space of dimension :math:`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).