Nil manifold

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


[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


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


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.


[a1] L. Auslander, "Lecture notes on nil-theta functions" , Amer. Math. Soc. (1977)
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Nil manifold. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article