$\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group?

Question

I try to prove $\operatorname{SL}(2,\mathbb R)$ is not isomorphic to $S^1 \times \mathbb R^2 $ as a Lie group.

My idea is that since $\exp\colon \mathfrak{sl}(2,\mathbb{R}) \to \operatorname{SL}(2,\mathbb{R})$ is not surjective, it is enough to show that $\exp\colon \operatorname{Lie}(S^1 \times \mathbb{R^2}) \to S^1 \times \mathbb{R^2}$ is surjective. But I couldn't find how to do it. Are there other good ideas? Could you help me?

Answer 1

$S^1 \times \Bbb{R}^2$ is abelian while $\text{SL}_2(\Bbb{R})$ is not.