Manifold

Conceptually an $n$-manifold is a space that looks like $\R^n$ locally. $\R^n$ is obviously a manifold. A sphere is a 2-manifold, as is a torus. The group SU(2) forms a 2-manifold. The Lorentz group SO(3,1) forms a 6-manifold.

More precisely, a manifold is a set $M$ with several subsets $U_i \subseteq M$ called charts. Each chart has a map $\phi_i : U_i \to \R^n$ that is invertible, smooth, and differentiable; and such that $\phi_i(U_i)$ forms an open region of $\R^n$. For any point $p \in M$, if $p$ is in several charts, they all map $p$ to the same value in $\R^n$.

A manifold has a vector space at each point called the tangent space. Together the tangent spaces form a tangent bundle. We can define a tangent space from the directional derivatives on the manifold. We say the tangent space at point $p$ consists of the set of directional derivatives along curves through $p$.

We can write down basis coordinate vectors $\vec e_{\mu} = \partial_\mu$ along the $x^\mu$ direction. It is easy to show these form a vector space and their composition also forms a partial derivative.