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I'm studying the chapter one of this text. In the end of the chapter there are a few exercises. The exercises 32 to 35 asks to constructs natural diffeomorphisms that maps fibers to fibers. The exercises are:

  1. $TS^1 \approx S^1 \times \mathbb R$.
  2. $T(M \times N) \approx TM \times TN$.
  3. $T\mathbb{R}^n \approx \mathbb R^n \times \mathbb R^n$.
  4. $TS^n \times \mathbb R \approx S^n \times \mathbb R^{n+1}$.

Where $TM$ denotes the tangent bundle of a smooth manifold $M$. I have no idea in how start to construct such diffeomorphisms. Can any one help me with at least one of this questions?

Thanks.

user 242964
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    For 1, see here. – A. Goodier Sep 19 '19 at 20:18
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    The 4th seems really weird: isn’t the second $\approx$ a $\times$ instead? If so, consider the sum of $TS^n$ and of the “tautological bundle” on $S^n$, ie you associate to any $x$ the line $\mathbb{R}x$. For the simpler 3., just choose $n$ global sections $s_1, \ldots, s_n$ of $T\mathbb{R}^n$ that are a basis at each point and the morphism is $(x,y) \in \mathbb{R}^n \times \mathbb{R}^n\times \sum_{k=1}^n{y_ks_k(x)}$. – Aphelli Sep 19 '19 at 20:51

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