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 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)