Nil manifold
From Encyclopedia of Mathematics
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 
of all matrices of the form
 |
and the discrete subgroup 
of all such matrices with integer 
, 
, 
. The corresponding quotient space 
of cosets 
, 
, 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://encyclopediaofmath.org/index.php?title=Nil_manifold&oldid=19191
Nil manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nil_manifold&oldid=19191
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article