3-4. The Tangent Bundle

Definition Tangent Bundle

• $M$,
  • smooth manifold with or without boundary

\[\begin{eqnarray} TM & := & \sqcup_{p \in M} T_{p}M \nonumber \\ & = & \sqcup_{p \in M} \{p\} \times T_{p}M \nonumber \\ & \cong & \sqcup_{p \in M} \{p\} \times \mathbb{R}^{n} \nonumber \\ & \cong & \mathbb{R}^{n} \times \mathbb{R}^{n} \end{eqnarray}\]

is called the tangent bundle of $M$. The tangent bundle comes equipped with a ntural projection map $\pi: TM \rightarrow M$

\[(p, v) \in TM, \ \pi((p, v)) = p .\]

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