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  "body": "\n## Definition\n\nLet $G$ be a connected [[solvable Lie group|9CNSVFQ801]] and let $H$ be a closed subgroup of $G$.\nThe [[homogeneous space|85TWR6E75P]] $G/H$, with its manifold structure as a quotient of [[Lie groups|7JBRGCVJRB]], is a *solvmanifold*.\nEquivalently, a solvmanifold is a manifold with a smooth transitive action of a connected solvable Lie group.\n\n## Remarks\n\nExample: $G$ the group of maps $x \\mapsto ax + b$ of $\\R$ with $a \u003e 0$ and $b \\in \\R$.\nIt is solvable, since its commutator subgroup lies in the abelian group of translations.\nIt acts transitively on $\\R$, and the stabilizer of $0$ is the closed subgroup $H = \\set{x \\mapsto ax}$.\nSo $\\R \\cong G/H$ is a solvmanifold.\n\nNon-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.\n\nMany authors call only the compact quotients $G/\\Gamma$ by a discrete subgroup $\\Gamma$ solvmanifolds.\nEach nilmanifold is a solvmanifold, since nilpotent groups are solvable.\n",
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