nLab Heisenberg manifold

Idea

A Heisenberg manifold is the quotient space of a real Heisenberg group by an integer Heisenberg subgroup.

By default this is understood in the case of the 3-dimensional Heisenberg group (which is the central group extension of the additive group $\mathbb{R}^2$ by the group cocycle which is the canonical symplectic form) and one refers to the Heisenberg manifold for this case, a 3-manifold.

Properties

Beware that Heisenberg manifolds does not inherit group structure since the integer Heisenberg group is not normal as a subgroup of the real Heisenberg group.

Proposition

The Heisenberg manifold is a nilmanifold.

Proposition

The Heisenberg manifold is a Seifert manifold.

References

See also:

• Yoshiaki Suzuki: Heisenberg Manifolds, section 2 of: The Folland-Stein spectrum of some Heisenberg Bieberbach manifolds, Kyushu Journal of Mathematics (2026) [arXiv:2402.15093]

• J. W. Cannon, W. J. Floyd, W. R. Parry; Ex. 7.4 in: A Survey of twisted face-pairing 3-manifolds [pdf]

• Richard Tolimieri: Analysis on the Heisenberg manifold, Amer. Math. Soc. 228 (1977) 329-343 [doi:10.1090/S0002-9947-1977-0447473-8, pdf]

Discussion in the context of sub-Riemannian geometry:

• Ovidiu Calin, Der-Chen Chang: Heisenberg Manifolds, Chapter 9 in: Sub-Riemannian Geometry – General Theory and Examples, Cambridge University Press (2009) 175–230 [doi:10.1017/CBO9781139195966.010]