I just read the definition of an immersed submanifold in Lee's book. My understanding of it is like a manifold that lives inside an other manifold but which are topologically not necessarily compatible. But I don't understand why we ask in the definition that the differential of the inclusion map to be injective at each point (i.e. $(d\iota)_p$ injective)? What is the idea behind this? why is this interesting?
2026-03-25 13:55:36.1774446936
Why do we ask the inclusion map to have full rank differential for a immersed submanifold?
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