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The point space $*$ is a smooth $0$-manifold.

I cannot determine if one should think that whenever $M$ is a smooth manifold, any map $M\to *$ is a smooth map of manifolds. I am not sure how to check the condition, since it would require me to write down a matrix with $0$ rows for the Jacobian (or check that all zero of the partial derivatives exist)? Does it hold vacuuously?

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  • $\begingroup$ It should indeed hold vacuously: any time you are checking if “all zero of something” exists, the answer is always yes. For instance, the statement “all rational numbers $a$ with $a^2=2$ also satisfy $a=a+1$” is true. The same reasoning applies here. $\endgroup$ – Danny Stoll Sep 30 '18 at 3:16
  • $\begingroup$ If it helps, think of the codomain as sitting inside $\Bbb R$: then your Jacobian is the zero matrix because you have a constant map. $\endgroup$ – user98602 Sep 30 '18 at 5:55

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