- #1

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I was trying to prove that the tangent bundle TM is a smooth manifold with a differentiable structure and I wanted to do it in a different way than the one used by my professor.

I used that TM=M x TpM. So, the question is:

**Can the tangent bundle TM be considered as the product manifold of a smooth manifold M and its tangent planes TpM with p∈M?.**

If this is the case, then can we conclude that it is a smooth manifold because it is a product of smooth manifolds and we can find a differentiable structure that comes from the maps of M and the maps of TpM?

If this is the case, then can we conclude that it is a smooth manifold because it is a product of smooth manifolds and we can find a differentiable structure that comes from the maps of M and the maps of TpM?

Thank you.