Homogeneous nilmanifolds with prescribed isotropy (Q1894606)

A homogeneous nilmanifold is a Riemannian manifold admitting a nilpotent transitive group of isometries \(N\). In this paper the following theorem is proved. Theorem: Let \(\mathfrak l\) be any compact Lie algebra. For every \(P \geq 2\) there exist a \(P\)-step nilpotent Lie group \(N\) and a left-invariant Riemannian metric \(g\) on \(N\) such that the Lie algebra of the isotropy group at the unity of \((N,g)\) is isomorphic to \(\mathfrak l\).