# Nil manifold

A compact quotient space of a connected nilpotent Lie group (cf. Lie group, nilpotent). (However, sometimes compactness is not required.)

#### References

[1] | A.I. Mal'tsev, "On a class of homogeneous spaces" Transl. Amer. Math. Soc. (1) , 9 (1962) pp. 276–307 Izv. Akad. Nauk SSSR Ser. Mat. , 13 : 1 (1949) pp. 9–32 |

#### Comments

Cf. also Nil flow and the references quoted there.

An example of a nil manifold that is rather important for various applications is the following. Consider the three-dimensional Heisenberg group $N$ of all matrices of the form

$$\begin{pmatrix}1&y&z\\0&1&x\\0&0&1\end{pmatrix}$$

and the discrete subgroup $\Gamma$ of all such matrices with integer $x$, $y$, $z$. The corresponding quotient space $\Gamma\setminus N$ of cosets $\Gamma n$, $n\in N$, is a compact nil manifold with an invariant probability measure. It plays an important role in harmonic analysis and the theory of theta-functions.

#### References

[a1] | L. Auslander, "Lecture notes on nil-theta functions" , Amer. Math. Soc. (1977) |

**How to Cite This Entry:**

Nil manifold.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Nil_manifold&oldid=32406