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I am studying the notion of tangent bundle $TM$ of a smooth manifold $M$,and in my textbook I am acknowledged that for most smooth manifold of dimension $n$,it's not true that $TM$ is diffeomorphic to $M \times \mathbb R^n$.I failed to give a simple example.I noticed that there is a set-theorical bijective correspondence between the to space,which makes it more difficult to understand why the diffeomorphism fails in general.Is there anyone to help?

Daniel Xu
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  • Have you learned about Euler class ? – Elad May 12 '19 at 07:41
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    https://math.stackexchange.com/questions/1323740/are-t-mathbbs-2-and-mathbbs-2-times-mathbbr2-different – Elad May 12 '19 at 08:36
  • Note that pure set-theoretic considerations is mostly useless in DG/DT context --- every manifold of positive dimension is going to have cardinality $c$ (because it is a countable union of $\mathbb{R}^n$). Problems are almost always in the topological aspect. – user10354138 May 12 '19 at 08:52

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