Solvmanifold

Remarks

Example: $G$ the group of maps $x \mapsto ax + b$ of $\R$ with $a > 0$ and $b \in \R$. It is solvable, since its commutator subgroup lies in the abelian group of translations. It acts transitively on $\R$, and the stabilizer of $0$ is the closed subgroup $H = \set{x \mapsto ax}$. So $\R \cong G/H$ is a solvmanifold.

Non-example: the presentation $S^2 = \mathrm{SO}(3)/\mathrm{SO}(2)$ does not show that $S^2$ is a solvmanifold, because $\mathrm{SO}(3)$ is not solvable: its Lie algebra $\mathfrak{so}(3)$ is simple.

Many authors call only the compact quotients $G/\Gamma$ by a discrete subgroup $\Gamma$ solvmanifolds. Each nilmanifold is a solvmanifold, since nilpotent groups are solvable.