The BTZ black hole as a Lorentz-flat geometry
  It is shown that 2+1 dimensional anti-de Sitter spacetimes are Lorentz-flat. This means, in particular, that any simply-connected patch of the BTZ black hole solution can be endowed with a Lorentz connection that is locally pure gauge. The result can be naturally extended to a wider class of black hole geometries and point particles in three-dimensional spacetime. 
Pedro D. Alvarez, Pablo Pais, Eduardo Rodriguez, Patricio   Salgado-Rebolledo and Jorge Zanelli
gr-qc hep-th