---- Let $M$ be a [[smooth manifold]] of dimension $n$. > [!definition] Definition. ([[tangent bundle]]) > The **tangent bundle of $M$**, denoted $TM$, is the [[disjoint union]] of the [[tangent space at a point of a smooth manifold|tangent spaces]] at all points of $M$: $TM = \coprod_{p \in M} T_{p}M.$ > Its elements are written $(p,v)$ for $p \in M$ and $v \in T_{p}M$. It is itself a [[smooth manifold]], of dimension $2n$. > $TM$ comes equipped with a natural projection map $\pi:TM \to M$ which sends each vector in $T_{p}M$ to the point $p$ at which it is tangent: $\pi(p,v)=p$. This makes it into a smooth [[vector bundle]] over $M$ — see below. > [!justification] $TM$ as a manifold. > Let $U$ be a [[coordinate chart|coordinate neighborhood]] for $M$, with coordinates $(x^{i})_{i=1}^{n}$ giving rise to [[tangent space to a manifold|coordinate bases]] $\left( \frac{ \partial }{ \partial x^{i} } |_{p}\right)_{i=1}^{n}$ for each $p \in U$. The **natural coordinates** on $TM$ are given by $\begin{align} \pi ^{-1}(U) &\to \mathbb{R}^{n} \times \mathbb{R}^{n} \\ \left( p, v^{i} \frac{ \partial }{ \partial x^{i} } |_{p} \right) & \mapsto \big( x^{1}(p), \dots, x^{n}(p) , v^{1}, \dots, v^{n}\big). \end{align}$ ^justification > [!justification] $TM$ as a vector bundle. > The natural coordinates give also a local trivialization $\begin{align} \Phi_{U}: \pi ^{-1}(U) &\to U \times \mathbb{R}^{n} \\ \left( p, v^{i} \frac{ \partial }{ \partial x^{i} } |_{p} \right) &\mapsto (p, v^{1}, \dots, v^{n}). \end{align}$ > (Or could do something like the the fiber-wise [[the Steenrod construction of a vector bundle over a smooth manifold|Steenrod construction]]) > [!basicexample] > - $T\mathbb{S}^{1}$ is [[diffeomorphism|diffeomorphic]] to $\mathbb{S}^{1} \times \mathbb{R}$. ^basic-example ---- #### ---- #### 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 > ```