---- > [!definition] Definition. ([[cotangent bundle]]) > Let $M$ be a [[smooth manifold]] of dimension $n$. The [[disjoint union topology|disjoint union]] $T^{*}M = \coprod_{p \in M} T_{p}^{*}M$is called the **cotangent bundle of $M$**. It comes with a natural projection map $\pi:T^{*}M \to M$ sending $\omega \in T_{p}^{*}M$ to the point $p \in M$ at which it is cotangent. > > Given any smooth [[coordinate chart|local coordinates]] $(x^{i})$ on an open subset $U \subset M$, for each $p \in U$ we denote the [[basis]] for $T_{p}^{*}M$ [[dual basis|dual]] to the [[tangent space at a point of a smooth manifold|coordinate basis]] $(\frac{ \partial }{ \partial x^{i} } |_{p})$ by $dx^{i} |_{p}$. This defines $n$ maps $dx^{1},\dots,dx^{n}:U \to T^{*}M$, called **coordinate covector fields**. > The **natural coordinates** $\begin{align} \pi ^{-1}(U) & \to \mathbb{R}^{n} \times \mathbb{R}^{n} \\ (p, \omega_{i} \ dx^{i} |_{p}) & \mapsto (x^{1}(p), \dots, x^{n}(p), \omega_{i}, \dots, \omega_{n} ) \end{align}$ turn $T^{*}M$ into a [[smooth manifold]] of dimension $2n$. A slight modification yields [[vector bundle|local trivializations]] turning $T^{*}M$ into a rank-$n$ [[vector bundle]] with typical fiber $\mathbb{R}^{n}$, via $\begin{align} \pi ^{-1}(U) & \to M \times \mathbb{R}^{n} \\ (p, \omega_{i} \ dx^{i} |_{p}) & \mapsto (p, \omega_{i}, \dots, \omega_{n} ) \end{align}$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```