Free actions of finite groups on the 3-dimensional nilmanifold

Torsion formulation of gravity

“…In fact, this is a sufficient condition for the existence of a pseudo-Riemannian metric of signature (p, q). Let U and U be local vielbeins of OL c (M) and OL c (M), respectively (see (2)). Due to the global splitting of the tangent bundle we can in fact choose in every open set…”

“…The resulting compact manifold is then parallelizable. It is in fact the twisted 3-torus [2]. This manifold is a non trivial T 2 -bundle over the base S 1 (described by the cyclic coordinate y).…”

“…Coming back to the 'reconstruction' of the global solution, it is surprising that in the new formulation different gluings are allowed, and topologies that could not be considered in the classical approach appear here as possible vacua. As a particularly interesting example we have the twisted torus [2], used in the literature as a compactification manifold of supergravity [3]. As we will see in example 3.3, the interpretation in this formalism is extremely easy and it could provide a simple way of proving that these compactifications are indeed spontaneous [4].…”