Nilmanifold
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In mathematics, the definition of a nilmanifold has not been universally agreed upon. One suggestion is the following: A nilmanifold is the quotient space of a nilpotent Lie group modulo a closed subgroup.
By this definition, a nilmanifold would then be a homogeneous space with a nilpotent Lie group acting transitively on it. However, this definition would not be consistent with the usage of the term compact nilmanifold. This article gives two finer definitions that are supported by the literature.
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[edit] Homogeneous nilmanifolds
The following definition comes from Wilson [1].
Let M be a connected Riemannian manifold and
- I(M)
the isometry group of M. M is said to be a homogeneous nilmanifold if I(M) contains a nilpotent Lie subgroup which acts transitively on M. It can then be shown (the main result of Wilson's paper) that M is isometric to N, where N is a nilpotent Lie group with left-invariant metric.
Another way to realize or construct these homogeneous nilmanifolds is to start with a simply connected nilpotent Lie group, and construct the quotient group by a discrete subgroup Γ lying in the centre of the group.
[edit] Compact nilmanifolds
A compact nilmanifold is a compact Riemannian manifold which is locally isometric to a nilpotent Lie group with left-invariant metric. These spaces are constructed as follows (see Raghunathan[2] for details).
Take a simply connected nilpotent Lie group N which admits a lattice. It is well known that a nilpotent Lie group admits a lattice if and only if its Lie algebra admits a basis with rational structure constants: this is Malcev's criterion. The lattice in the Lie algebra gives rise to a discrete subgroup Γ. We endow N with a left-invariant (Riemannian) metric. Now Γ can be view as a discrete group of isometries acting on N by left mutiplication, since we endowed N with a left-invariant metric.
To construct the compact nilmanifold we quotient N by the group action of Γ and obtain 
. Note this is not the same space as N / Γ viewed as a homogeneous space.
[edit] Examples
[edit] Nilpotent Lie groups
From the above definition of homogeneous nilmanifolds, it is clear that any nilpotent Lie group with left-invariant metric is a homogeneous nilmanifold. The most familiar nilpotent Lie groups are matrix groups whose diagonal entries are 1 and whose lower diagonal entries are all zeros.
E.g., the Heisenberg group is a nilpotent Lie group. This nilpotent Lie group is also special in that it admits a compact quotient. The group Γ would be the upper triangular matrices with integral coefficents. Both of these nilmanifolds are 3-dimensional.
[edit] Abelian Lie groups
A simpler example would be any abelian Lie group. This is because any such group is a nilpotent Lie group. A familiar example might be the compact 2-torus or Euclidean space under addition.
[edit] Generalizations
A parallel construction based on solvable Lie groups produces a class of spaces called solvmanifolds.
