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  "body": "\n## Definition\n\nLet $N$ be a connected [[nilpotent Lie group|DK87YN458D]] and let $H$ be a closed subgroup of $N$.\nThe [[homogeneous space|85TWR6E75P]] $N/H$ is a *nilmanifold*.\nWhen $N$ is simply connected and $H = \\Gamma$ is a [[lattice|BTYF5Y5PFY]], a discrete subgroup with $N/\\Gamma$ compact, the space $N/\\Gamma$ is a *compact nilmanifold*.\n\n## Remarks\n\nExample: the [[Heisenberg group|ZZYCV5EX9J]] $N$ of real matrices $\\begin{pmatrix} 1 \u0026 x \u0026 z \\\\ 0 \u0026 1 \u0026 y \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}$ and the subgroup $\\Gamma$ of those with $x, y, z \\in \\Z$.\nThe group $N$ is nilpotent, since its commutator subgroup is the center $\\set{x = y = 0}$.\nIt is diffeomorphic to $\\R^3$, so it is simply connected.\nThe subgroup $\\Gamma$ is discrete, and each element of $N$ is an element with $x, y, z \\in [0, 1)$ times an element of $\\Gamma$, so $N/\\Gamma$ is compact.\n\nNon-example for the compact case: let $\\Z$ be the integer points on the first axis of $\\R^2$.\nThen $\\R^2/\\Z$ is a nilmanifold, but $\\Z$ is not a lattice in $\\R^2$, and the cylinder $\\R^2/\\Z$ is not compact.\n\nEach nilmanifold is a [[solvmanifold|HBDEPTRF11]].\nMany texts use the word nilmanifold only for compact nilmanifolds.\nBy a theorem of Malcev, a simply connected nilpotent Lie group has a lattice exactly when its Lie algebra has a basis with rational structure constants.\n",
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