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If$ f:X\to Y$ is an Immersion\ Submersion\ Diffeomorphism then what can we say about the induced map $ df: T(X)\to T(Y) $ defined as $df(x,v)= (f(x),df_x(v))$. Will it also have the same properties? I don't know how to start, kindly help.

Thanks & regards

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Hint: The best way to see this is to look it locally. If $f$ is a diffeomorphism, $f^{-1}$ is a diffeomorphism and the inverse of $df$ is $df^{-1}$ defined by $df^{-1}(x,v)=(f^{-1}(x),df^{-1}(v))$.

If $f$ is an immersion submersion and$x\in X$ there exists a chart $U$ containing $x$ such that the restriction of $f$ to $U$ is an immersion/submersion whose image is contained in a chart containing $f(y)$. It is enough to show the result when $f:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^m$. In this case, immersion submersion are characterized by the constant rank theorem and local diffeomorphism

see

Local Submersion Theorem - Differential Topology of Guillemin and Pollack

and

Local Immersion Theorem in $\mathbb{R}^n$ proof

which reduce the problem to a linear map.

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    $\begingroup$ Study the local situation with the local inversion theorem. Remark that the tangent space of an open subset is a trivial bundle. $\endgroup$ – Tsemo Aristide Sep 8 at 13:59
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    $\begingroup$ smoothness comes for free in this case. $\endgroup$ – Anubhav Mukherjee Sep 8 at 21:41
  • $\begingroup$ But How, I am talking about the Derivative map , the second argument in the ordered pair of df @AnubhavMukherjee $\endgroup$ – Devendra Singh Rana Sep 9 at 3:40
  • $\begingroup$ Yeah, I know that tangent space of open subset is actually the whole Tangent space. Does this helps with the smoothness of $df_x $@TsemoAristide ? $\endgroup$ – Devendra Singh Rana Sep 9 at 8:49

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