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I want to prove that the tangent bundle of open annulus is diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^3$.

This arguments came from mathoverflow

I have no clue of constructing this, any rudimental information will be helpful.

I have some basic information of constructing $T\mathbb{S}^1 $ is diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^1%$.

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The result can be deduced from the following facts.

(1) The annulus is diffeomorphic to $S^1\times R^1$.

(2) The tangent bundle of $S^1$ is $S^1\times R^1$. See the following post: Tangent bundle of $S^1$ is diffeomorphic to the cylinder $S^1\times\Bbb{R}$. The tangent bundle of $R^1$ is obviously $R^2$.

(3) The tangent bundle of a product manifold $M_1\times M_2$ is $TM_1\times TM_2$. Here $TM$ denotes the tangent bundle of $M$.

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