Can the Lie group structure be recovered from the geometry of an invariant metric?

Question

Is there a manifold $M$ with two non isomorphic Lie group structures $G_{1}$ and $G_{2}$, and two left invariant metrics $g_{1}$ and $g_{2}$, respectively such that $(M,g_{1})$ is isometric to $(M, g_{2})$?

Answer 1

Here you have an example in the Riemannian case. Take $M=\mathbb{R}^3$ as smooth manifold. Then $M$ is diffeomorphic to the the universal covering of the Lie group $E(2)$ of rigid motions of the Euclidean $2$-space. So we can consider $M$ with two different non isomorphic Lie group structure. Namely, the above one and the obvious one as $3$-dimensional abelian group. By Corollary 4.8 at page 309 in Milnor's paper, there is a flat metric $g$ on $E(2)$. So $(M,g)$ and $(M,g_0)$ are isometric, where $g_0$ is the standard flat metric but $E(2)$ and $\mathbb{R}^3$ regarded as Lie groups are not isomorphic.