I am learning the theory of smooth manifolds and have a question on the definitions of a submersion/immersion and its dependency on given charts.
Given a smooth map $f:M\mapsto N$ between two smooth manifolds of finite dimension. If I am correct this means that given any chart $\chi$ of $M$ and chart $\chi^\prime$ on $N$,
$$f_{\chi^\prime}^{\chi}=\chi^\prime\circ f\circ\chi^{-1}$$
is smooth in the usual sense of analysis.
Now to prove if $f$ is a submersion (or similar an immersion) at $p\in M$ one checks that,
$$(df_{\chi^\prime}^{\chi})_{\chi(p)}$$
is surjective\injective. By the chainrule,
$$\big(d(\chi^\prime\circ f\circ\chi^{-1})\big)_{\chi(p)}=(d\chi^\prime)_{\chi(p)}\circ(df)_p\circ(d\chi^{-1})_{\chi^{-1}(p)}.$$
But since all charts a homeomorphisms their differentials are isomorfisms.
Now my question is, why bother looking at $f_{\chi^\prime}^{\chi}$ if you can just look at whether or not the differential of $f$ is surjective/injecitive? The differentials of the charts are after all isomorfisms. Am i looking at it the right way?
Beside that, in the practical situation of having to check wheter or not a map is a submersion/immersion one has to do this for all combination of charts contained in the two atlases which induce the smooth structures, thats a bit cumbersome… Is there a trick/theorem one can use?
Best Answer
To put it simply, you can. You ise an atlas to define a smooth structure, so the fact that $f:M\to N$ is smooth is in part determined by the coordinate atlases, but for any smooth function, $df_p$ is a coordinate-independent linear map from $T_pM\to T_{f(p)}N$, and immersions/submersions are typically defined in terms of the rank of $df$ without reference to coordinates.
Specifically, we say $f$ is
• an immersion if $df_p$ is injective at each $p\in m$
• a submersion if $df_p$ is surjective at each $p\in m$
Of course, if one wants to compute anything on a concrete manifold, it generally requires using coordinates, so characterizing submersions/immersions in terms of coordinates is still useful.
As you have shown, the rank of $df$ is equal to the rank of its local representative in any coordinates, so we can check if a map is a submersion/immersion in coordinates without issue. There's no need to use every combination of charts, only enough to cover each point in $M$ at least once.