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I want to show that there's no immersion from $S^n$ to $\mathbb R^n$, that is, there exists no smooth function $f:S^n\rightarrow \mathbb R^n$ such that $df|_p$ is injective $\forall p \in S^n$.

By simple calculation, we know that it's equivalent to say that after writing $f = (f_1,f_2,\cdots, f_n)$, the matrix $(\frac{\partial f_i}{\partial x_j}|_p)_{i,j}$ has determinant $\neq 0\, \forall p \in S^n$ where $x_i$ are local coords of $S^n.$

But I don't know what to do next.

Thomas Andrews
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CYC
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  • Related https://math.stackexchange.com/questions/1832258/show-that-there-exists-no-immersion-f-of-s-1-into-r1 – Edu Oct 04 '17 at 16:09

1 Answers1

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An immersion would be an open mapping, so its image would be both open and compact...

Angina Seng
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  • Is every immersion an open mapping? This answer https://math.stackexchange.com/a/436573/360587 suggests not. (The counter example is the canonical immersion from R^1 into R^2. The domain is open but the image is a line in R^2 which is not open.) – Aryaman Jal Mar 18 '19 at 16:18
  • An immersion between manifolds of the same dimension is an open map. @AryamanJal – Angina Seng Mar 18 '19 at 17:41
  • Oh sorry, of course you meant same dimension in the answer above. Because such immersions would be in fact be local diffeomorphisms and in particular submersions which we know are open. – Aryaman Jal Mar 18 '19 at 17:57