Let \(M\) be a smooth manifold. Then the tangent bundle is defined as the disjoint union of all tangent spaces on \(M\), i.e.

\begin{equation*} TM=\coprod_{p \in M} T_pM. \end{equation*}
Remarks
  • \(TM\) is a vector fiber bundle over \(M\)
  • If \(M\) is a [smooth \(n\)-manifold](smooth manifold.md), then \(TM\) is a smooth \(2n\)-manifold [1, Proposition 3.18].

See also Link to heading

References Link to heading

  1. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.