Low dimensional Lie groups admitting left invariant flat projective or affine structures (Q417648)

A left invariant flat affine structure is a torsion free affine connection \(\nabla\) on a Lie group \(L\) such that \(\nabla\) is left invariant and flat. A left invariant flat projective structure \(\left|\nabla\right|\) is a projective equivalence class of an affine connection \(\nabla\) on \(L\) such that the left action of \(L\) is a projective transformation and \(\nabla\) is locally projectively equivalent to some flat affine connection. The author proves that any real Lie group of dimension \(\leq 5\) admits a left invariant flat projective structure. He also proves that a real Lie group \(L\) of dimension \(\leq 5\) admits a left invariant flat affine structure if and only if the Lie algebra of \(L\) is not perfect.