I was wondering whether any immersion is locally injective?-This definitely sounds natural and I guess that it is true, as the derivative of an immersion is globally injective. Thus, there cannot be an area where our immersion is constant. Am I correct about it?-Probably this would imply that we can write any immersion locally as a graph of a function, right?
2026-04-04 02:25:35.1775269535
Immersion locally injective?
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You can use the Inverse Function Theorem to prove that, in suitable local coordinates on both manifolds, an immersion is locally of the form $$f(x_1,\dots,x_k)=(x_1,\dots,x_k,0,\dots,0).$$