A $C^{\infty}$-manifold, which is not a $C$-analytic manifold; and a $C$-analytic manifold, which is not a $C^{\infty}$-manifold.

Question

My Question: I am looking for two counter-examples:

A $C^{\infty}$-manifold, which is not a $C$-analytic manifold.
A $C$-analytic manifold, which is not a $C^{\infty}$-manifold.

I think a lot, but I have no idea.

Why do I ask this question?

Motivating by this question $C^n$-manifold, which is not a $C^{n+1}$-manifold, I think about two natural questions:

Is every $C^{\infty}$-manifold, is a $C$-analytic manifold?
Is every $C$-analytic manifold, is a $C^{\infty}$-manifold?

I think the answer to these two questions should be negative, because otherwise there should be some famous propositions about these very natural questions, but I didn't find anything.

Two simpler questions you may want to consider: Is every smooth function analytic? Is every analytic function smooth? — Kajelad, Nov 04 '20 at 17:10
@Kajelad, I am not good at these topics. And I can not get the points just by some hints. — Tireless and hardworking, Nov 04 '20 at 17:14

score 3 · Accepted Answer · answered Nov 04 '20 at 17:16

Analytic functions are $C^\infty$, so an analytic atlas is automatically a $C^\infty$ atlas. So analytic manifolds are always $C^\infty$. The reverse is not true, and a counterexample is easy to find: take any non-analytic $C^\infty$-diffeomorphism $\mathbb R\to\mathbb R$, and take a compatible $C^\infty$-atlas of $\mathbb R$ containing the identity map. You get a $C^\infty$-manifold which is not analytic.

This is not a famous proposition because it isn't particularly challenging to prove, and also not a very deep result.

A $C^{\infty}$-manifold, which is not a $C$-analytic manifold; and a $C$-analytic manifold, which is not a $C^{\infty}$-manifold.

1 Answers1