Definition: Tangent Bundle

Let \(M\) be a differentiable manifold. The set. \[TM=\biguplus _{x\in M}T_{x}M,\]

together with the projection function

\[\pi \colon TM\longrightarrow M,\,(x,v)\longmapsto x\,,\]

is called the tangent bundle of \(M\). So, an element of \(TM\) can be thought of as a pair \((x,v)\), where \(x\) is a point in \(M\) and \(v\) is a tangent vector to \(M\) at \(x\).

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück