Classification of solutions in topologically massive gravity

We study exact solutions of three-dimensional gravity with a cosmological constant and a gravitational Chern-Simons term: the theory known as topologically massive gravity. After reviewing the algebraic classification, we show that if a solution has curvature of algebraic type D, then it is biaxially squashed AdS_3. Applying the classification, we provide a comprehensive review of the literature, showing that most known solutions are locally equivalent to biaxially squashed AdS_3 or to AdS pp-waves.Comment: 39 pages. v1 split into 2 separate papers. This version is the first of the two, containing the classification of solutions, with a strengthening of the type D result. The material on Kundt solutions now appears in the second paper arXiv:0912.3438. v3: minor changes