Let $M$ be a [smooth manifold](/page/Smooth%20Manifold) of dimension $n$, and let $TM = \bigsqcup_{p \in M} T_pM$ be its tangent bundle with projection $\pi: TM \to M$ given by $\pi(\xi) = p$ for $\xi \in T_pM$. Then $TM$ admits a canonical [smooth vector bundle](/page/Smooth%20Vector%20Bundle) structure over $M$ of rank $n$, whose fiber over each $p \in M$ is the tangent space $T_pM$.