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- TL;DR Summary
- How one show isomorphism?

Hello there!

Reading the textbook on differential geometry I didn't get the commentary. In Chapter about vector bundles authors provide the following example

Let ##M=S^1## be realized as the unit circle in ##\mathbb{R}^2##. For every ##x\in S^1##, the tangent space ##T_x S^1## can be identified with subspace of vectors orthogonal to ##x##. This yields a bijection ##\Phi## from ##TS^1## onto subset

##T=\{ (x,X)\in S^1 \times \mathbb{R}^2\, :\, x \perp X \}## of ##\mathbb{R}^4##.

This is the level set of the smooth mapping

##F:\mathbb{R}^4 \rightarrow \mathbb{R}^2\, ,\;\; F(x,X):=( \| x \|^2,\; x\cdot X)##

at the regular value ##c=(1,0)##. Hence it carries a smooth structure. One can check that ##\Phi## is a diffeomorphism with respect to this structure.

Diffeomorphism in this particular example seems completely tautological however I don't get this constructions of charts with ##pr_k##. I understand how one can build the charts on ##S^1## using projections ##pr_k## nonetheless how this should help with diffeomorphism proof is not clear.

Reading the textbook on differential geometry I didn't get the commentary. In Chapter about vector bundles authors provide the following example

Let ##M=S^1## be realized as the unit circle in ##\mathbb{R}^2##. For every ##x\in S^1##, the tangent space ##T_x S^1## can be identified with subspace of vectors orthogonal to ##x##. This yields a bijection ##\Phi## from ##TS^1## onto subset

##T=\{ (x,X)\in S^1 \times \mathbb{R}^2\, :\, x \perp X \}## of ##\mathbb{R}^4##.

This is the level set of the smooth mapping

##F:\mathbb{R}^4 \rightarrow \mathbb{R}^2\, ,\;\; F(x,X):=( \| x \|^2,\; x\cdot X)##

at the regular value ##c=(1,0)##. Hence it carries a smooth structure. One can check that ##\Phi## is a diffeomorphism with respect to this structure.

**(To see this, let ##pr_k:\mathbb{R}^2 \rightarrow \mathbb{R}## denote the natural projection to the ##k##-th component and choose the charts on ##S^1## and ##T## to be restrictions of ##pr_k## and ##pr_k \times pr_k##, respectively, ##k=1,2##.)**Thus, ##TS^1## can naturally be identified with ##T##.Diffeomorphism in this particular example seems completely tautological however I don't get this constructions of charts with ##pr_k##. I understand how one can build the charts on ##S^1## using projections ##pr_k## nonetheless how this should help with diffeomorphism proof is not clear.