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Show that $f:(x_1,...,x_k)\in\mathbb S^k\mapsto(x_1,...,x_k,0,...,0)\in\mathbb S^n$ is an embedding.

I am new to differential geometry, and I have a hard time wrapping my head around this one. The answer should be really trivial but I can't quite find how to prove this. We must show that $f$ is an immersion and a homeomorphism onto $f(\mathbb S^k)$.The latter is very straight forward, but to prove the first point I must show that $d(\psi\circ f\circ\phi^{-1})_a$ has rank $k$ for all charts $\phi$ at $a\in\mathbb S^k$ and $\psi$ at $f(a)\in\mathbb S^n$.

I do apologize if the answer is trivial, and I thank anyone who can bring me help.

maxjw91
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  • Just work in local coordinates here. You have $n$ angles for the $n$-sphere and $k$ angles for the $k$-sphere. Find out how the spherical coordinates are related and you basically have your answer, since you will end up with a matrix consisting of a $k \times k$ unit matrix and $0$ otherwise.
    I will leave this here as a starter: https://en.wikipedia.org/wiki/N-sphere     – F. Conrad Sep 20 '22 at 11:16
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    Alternatively, you can use the fact that this map is the restriction and the corestriction of the natural embedding $\Bbb R^k \to \Bbb R^n$, and use the fact that the differential of the restriction to a submanifold is the restriction of the differential to the tangent space of that submanifold: there you should see easily that $f$ is an immersion – Didier Sep 20 '22 at 12:58
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    The embedding should be $\mathbb R^{k+1}$ to $\mathbb R^{n+1}$ :) – Sven-Ole Behrend Sep 20 '22 at 13:03
  • @Sven-OleBehrend Of course you're right! Too late to edit unfortunately – Didier Sep 20 '22 at 14:21

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