Cotangent bundle

Let $M$ be a smooth manifold of dimension $n$. For each $p\in M$, the tangent space $T_pM$ is a real vector space, and its dual space

is the cotangent space at $p$.

Definition (cotangent bundle). The cotangent bundle of $M$ is the disjoint union

together with the projection map $\pi:T^*M\to M$ sending a covector $\alpha\in T_p^*M$ to its base point $p$.

Smooth structure / vector bundle structure. The set $T^*M$ carries a canonical smooth manifold structure of dimension $2n$ such that:

1. $\pi:T^*M\to M$ is a smooth map, and each fiber $\pi^{-1}(p)=T_p^*M$ is a vector space of dimension $n$.
2. For every smooth chart $(U,x)$ on $M$, with coordinates $x=(x^1,\dots,x^n)$, there is a smooth trivialization
   $$\Phi_U:\pi^{-1}(U)\to U\times \mathbb{R}^n$$
   defined as follows: each $\alpha\in T_p^*M$ (with $p\in U$) can be written uniquely as
   $$\alpha=\sum_{i=1}^n a_i\, (dx^i)_p,$$
   and then $\Phi_U(\alpha)=(p,(a_1,\dots,a_n))$.
3. On overlaps $U\cap V$, the induced transition functions are smooth and linear in the fiber variables, so $\pi:T^*M\to M$ is a smooth vector bundle of rank $n$ (the dual bundle of the tangent bundle).

Smooth sections of $T^*M$ are exactly differential $1$-forms, i.e. the case $k=1$ of a differential $k$-form.

1. Euclidean space. For $M=\mathbb{R}^n$ with standard coordinates, each $T_p\mathbb{R}^n\cong \mathbb{R}^n$ canonically, hence $T_p^*\mathbb{R}^n\cong (\mathbb{R}^n)^*$. Using the standard basis $dx^1,\dots,dx^n$, one gets a global trivialization
   $$T^*\mathbb{R}^n \cong \mathbb{R}^n\times (\mathbb{R}^n)^* \cong \mathbb{R}^{2n}.$$

1. The circle. For $M=S^1$, the cotangent bundle is a rank-$1$ vector bundle and is trivial:
   $$T^*S^1 \cong S^1\times \mathbb{R}.$$
   Concretely, on any local angular chart, every covector at $p\in S^1$ is of the form $a\, d\theta|_p$ for a unique $a\in\mathbb{R}$. These local descriptions patch to the global trivialization; there is no single-valued global angular coordinate on $S^1$.

1. Lie groups. If $G$ is a Lie group, then left translation canonically identifies each $T_gG$ with $T_eG$. Dualizing gives the left trivialization
   $$T^*G\cong G\times\mathfrak g^*,$$
   where $\mathfrak g=T_eG$ is the Lie algebra of $G$.