Jonginn Yun

Manifolds

A smooth $n$-manifold is a space locally modeled on $\mathbb R^n$ with smooth transition maps between coordinate charts. A chart is a map

$$\varphi:U\to \varphi(U)\subset \mathbb R^n,$$

where $U$ is open in the manifold.

The point of the definition is not that the manifold is secretly $\mathbb R^n$. It is that calculus can be done locally, and the answers can be checked to transform consistently when coordinates change.

Tangent Vectors

At a point $p\in M$, a tangent vector can be understood as a directional derivative acting on smooth functions:

$$v:C^\infty(M)\to \mathbb R.$$

It is linear and satisfies the Leibniz rule

$$v(fg)=v(f)g(p)+f(p)v(g).$$

In local coordinates $x^1,\ldots,x^n$, every tangent vector can be written

$$v=v^i\frac{\partial}{\partial x^i}\bigg|_p.$$

The coordinate expression changes under a chart change, but the tangent vector itself is intrinsic.

Cotangent Vectors

The cotangent space $T_p^*M$ is the dual vector space of $T_pM$. Its elements are covectors. In coordinates, the dual basis is

$$dx^1,\ldots,dx^n,$$

with

$$dx^i\left(\frac{\partial}{\partial x^j}\right)=\delta^i_j.$$

A one-form is a smoothly varying choice of cotangent vector:

$$\alpha=\alpha_i dx^i.$$

This language is the beginning of differential forms and integration on manifolds.

Bundles Already Appear

The tangent spaces over all points assemble into the tangent bundle

$$TM=\bigsqcup_{p\in M}T_pM.$$

Vector fields are sections of $TM$, and one-forms are sections of $T^*M$. This is why differential geometry naturally leads into vector bundles.