Why is it that a smooth homomorphism of lie groups is either an immersion or submersion?
Is this obvious? I'm not seeing it at the moment. Any help is much appreciated!
2026-05-06 00:16:15.1778026575
Why is it that a smooth homomorphism of lie groups is either an immersion or submersion?
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Strictly speaking, this isn't true. Consider any linear map $V \to W$ of real vector spaces. This is certainly a smooth homomorphism of lie groups, and it is its own differential at the origin under standard identifications. So just pick a map that's neither injective nor surjective, and then it won't be an immersion or a submersion at any point in the kernel.
However, it is true that any smooth homomorphism is a submersion onto its image (equivalently, can be written by a submersion, followed by an immersion, where both pieces are also smooth homomorphisms). To see this (maybe this is overkill) note that by Sard's theorem there is a non-critical point, and then because $G$ moves fibers to fibers via diffeomorphisms, there are no critical points.