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This seems obvious, but I'm having trouble carrying through the details.

Suppose there is a smooth function $f$ with zero derivative on a manifold $M$ with $n$ connected components. Why is $f$ constant on each connected component?

Detailed answers are very much appreciated. Thanks!

Potato
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    What exactly do you mean by derivative here? – Mariano Suárez-Álvarez Jan 07 '12 at 05:58
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    I suggest assuming $n=1$. Do you know how to prove it when $M=\mathbb R^k$? – Jonas Meyer Jan 07 '12 at 06:00
  • @MarianoSuárez-Alvarez I guess that's part of my question. I only know how to prove this when $M=\mathbb{R}$, and I don't know enough differential geometry to define derivatives on manifolds. But wikipedia claims this is true. – Potato Jan 07 '12 at 06:15
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    But then you should probably pick a textbook dealing with the subject and learn that first! It is extraordinarily understandable that you be having problems with proving this if you do not know what derivatives are in this context. – Mariano Suárez-Álvarez Jan 07 '12 at 06:17
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    (The only way to detailedly answer this starting from what a derivative is to the claim you want to prove is to more or less write out an exposition of what a manifold is and what a smooth function on it is: this is not the best way to use this site) – Mariano Suárez-Álvarez Jan 07 '12 at 06:19
  • @MarianoSuárez-Alvarez Indeed. I was hoping that once I saw a proof with the proper definition, I could follow along. – Potato Jan 07 '12 at 06:19
  • @MarianoSuárez-Alvarez I know what a manifold is, and how to define functions on them. I'm a little fuzzy on derivative taking, that is all. – Potato Jan 07 '12 at 06:20

1 Answers1

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Let $M$ be an $m$-manifold. We'll concentrate on a connected component of $M$, say $U$. Pick $p\in U$, let $V_p$ be a neighborhood of $p$ in $U$ admitting a local Euclidean chart, and let $\phi: D\subset\mathbb{R}^m\rightarrow V_p$ be a coordinate chart. If $f: M\rightarrow \mathbb{R}$ is a differentiable function on $M$, this really means that $f\circ \phi: \mathbb{R}^m\rightarrow \mathbb{R}$ is a differentiable function (in fact, this is the definition of a differentiable function on $M$). Now prove that $f\circ \phi$ is constant using standard calculus. So $f$ is constant on $V_p$. From the fact that $U$ is connected, conclude by standard topological arguments that $f$ is constant on $U$.

I would suggest picking up a book on differential geometry and topology.

  • This is crystal clear. Thank you! – Potato Jan 07 '12 at 07:09
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    One wonders how you followed this along, being that it does not contain a definition of the derivative you mention in your question (in fact, it does not even mention the hypothesis! :D ) – Mariano Suárez-Álvarez Jan 07 '12 at 09:41
  • Is there something unclear about the differential of a map from $\mathbb{R}^m$ to $\mathbb{R}^n$? Presumably he does know how to differentiate multivariable maps. If not, then he should pick up a book on elementary calculus, before looking to geometry (my answer does make a reference to calculus, so he knows where to look :-D). –  Jan 07 '12 at 10:02
  • @MarianoSuárez-Alvarez What it means to be a differentiable function on a manifold is defined in the answer. – Potato Jan 07 '12 at 22:51
  • @WNY Do not worry. I am familiar with multivariable analysis, just not so much with differential geometry. – Potato Jan 07 '12 at 22:52
  • What if the function is between two manifolds? – user118413 Jun 08 '21 at 18:47