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  "body": "\n## Summary\n\nA nilmanifold is a homogeneous space $N/H$ of a connected [[nilpotent Lie group|DK87YN458D]] $N$ by a closed subgroup $H$.\nThe most studied ones are the compact quotients $N/\\Gamma$ of a simply connected nilpotent Lie group by a lattice $\\Gamma$.\nThe torus is the abelian case, and the Heisenberg manifold is the first nonabelian one.\nCompact nilmanifolds give many examples in geometry, such as symplectic manifolds that are not Kähler.\n\n## Explanations\n\n### L1 human\n\nThe torus is the plane $\\R^2$ folded up by the integer translations.\nA nilmanifold does the same with a group that is almost commutative, a nilpotent group.\nThe main example uses upper triangular $3 \\times 3$ matrices with ones on the diagonal and real entries above it.\nFolding them up by the matrices with integer entries gives a compact space of dimension $3$, the Heisenberg manifold.\nIt looks like a torus bundle that twists as one goes around a circle.\n",
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