# Smooth map to the $0$-manifold?

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?

• 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. – Danny Stoll Sep 30 '18 at 3:16
• 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. – user98602 Sep 30 '18 at 5:55