Low dimensional Lie groups admitting left invariant flat projective or   affine structures
  We prove that any real Lie group of dimension \leq 5 admits a left invariant flat projective structure. We also prove 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. 
Hironao Kato
math.DG math.RT