---
tag: 43NRBE72C6
kind: concept
name: Nilmanifold
msc: ["22E25", 22F30, 53C30]
specializes: [HBDEPTRF11]
sources:
  - url: https://en.wikipedia.org/wiki/Nilmanifold
same_as:
  wikidata: Q4321368
  wikipedia: Nilmanifold
---

## Summary

A nilmanifold is a homogeneous space $N/H$ of a connected [[nilpotent Lie group|DK87YN458D]] $N$ by a closed subgroup $H$.
The most studied ones are the compact quotients $N/\Gamma$ of a simply connected nilpotent Lie group by a lattice $\Gamma$.
The torus is the abelian case, and the Heisenberg manifold is the first nonabelian one.
Compact nilmanifolds give many examples in geometry, such as symplectic manifolds that are not Kähler.

## Explanations

### L1 human

The torus is the plane $\R^2$ folded up by the integer translations.
A nilmanifold does the same with a group that is almost commutative, a nilpotent group.
The main example uses upper triangular $3 \times 3$ matrices with ones on the diagonal and real entries above it.
Folding them up by the matrices with integer entries gives a compact space of dimension $3$, the Heisenberg manifold.
It looks like a torus bundle that twists as one goes around a circle.