# Is a smooth Lie group an analytic Lie group?

Maybe related : Why are Lie groups automatically analytic manifolds?

I'm seraching for a precise statement. I propose two theorems.

Theorem 1

Let $$G$$ be a smooth manifold (the transitions functions are $$C^{\infty}$$ as maps between open subsets of $$R^n$$). We suppose that $$G$$ is a group for which the mulliplication and the inverse are smooth.

Then the transition functions, the multiplication and the inverse are analytic.

Theorem 2

Let $$G$$ be a smooth manifold (the transitions functions are $$C^{\infty}$$ as maps between open subsets of $$R^n$$). We suppose that $$G$$ is a group for which the mulliplication and the inverse are smooth.

Then $$G$$ accepts an analytic atlas for which the multiplication and the inverse are analytic. Moreover every two such atlas are analytically diffeomorphic.

My guesses:

• Theorem 1 is wrong because the atlas may contain smooth charts which are not analytic.
• Theorem 2 is true.

QUESTIONS:

• is theorem 1 wrong ?
• is theorem 2 true ?
• can I have a link (online; I do not have access to books) to a precise statement and a proof ?

My aim is to be sure that I can always suppose every Lie groups to be analytic (transition charts, multiplication, inverse) without loss of generality.

Consider the following charts: around $$e$$ it's $$X \mapsto \exp X$$, from a neighborhood of $$0$$ in $$\frak{g}$$ to a neighborhood of $$e$$ in $$G$$. Around a $$g\in G$$ it's $$X \mapsto g \exp X$$. This give an analytic atlas in which multiplication and inverse is analytic. An argument for this: say we have the chart aroung a $$g= \exp X_0$$ close to $$e$$, $$X \mapsto \exp X_0 \exp X$$. To see the point $$\exp X_0 \exp X$$ in the chart around $$e$$ we need to write the equality $$\exp X_0 \exp X = \exp X_1$$ Now, $$X_1$$ can be expressed in terms of $$X_0$$ and $$X$$ using the B-C-H formula.
• @Laurent Claessens: The conclusion is that a $C^{\infty}$ Lie group "is" an analytic Lie group in a very precise sense -- unique also. This can be proved similarly for Lie group of class $C^{3}$ (and more difficult but still true for lower class, up to $C^0$). Commented Aug 24, 2021 at 18:21