Induced map on $T(X)$ the Tangent bundle

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

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.

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