From Wikipedia:
Submersions:
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.
Immersions:
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective.
For submersions, I was wondering if the differentiable map between differentiable manifolds will always hold as a diffeomorphism. I would suspect that:
(i) Some submersions are diffeomorphisms, but not all. Surjectivity doesn't imply bijectivity, which means that the map may not always be invertible, thus making it not always a diffeomorphism.
Is this correct? Likewise,
(ii) Some immersions are diffeomorphisms, but not all, since injectivity doesn't imply bijectivity either. This is nicely dual to the subjectivity of submersions. To emphasize, this map may not always be invertible, making it sometimes but not always a diffeomorphism.
Is this also correct? Thanks in advance!