ABSTRACT

This formal correspondencemotivates yet another explanation of the locallyflat nature of the string metric (7.28). In 2 + 1 dimensions both the Ricci tensor and the Riemann tensor have only six algebraically-independent components (in contrast to the situation in 3 + 1 gravity, where the Ricci tensor has 10 algebraically-independent components and the Riemann tensor 20). Thus, whenever the Ricci tensor vanishes the Riemann tensor automatically vanishes as well. In other words, spacetime is locally flat in any vacuum region in 2 + 1 dimensions. This property is inherited also by the string metric (7.28).