Dilatonic Parallelizable NS-NS Backgrounds

We complete the classification of parallelizable NS-NS backgrounds in type II super-gravity by adding the dilatonic case to the result of Figueroa-O'Farrill on the non-dilatonic case. We also study the supersymmetry of these parallelizable backgrounds. It is shown that all the dilatonic parallelizable backgrounds have sixteen supersymmetries. The parallelizable pp-wave backgrounds have been recently discussed in [1]. They are a class of the non-dilatonic NS-NS backgrounds in type II supergravity. Immediately, Figueroa-O'Farrill has classified all the parallelizable NS-NS backgrounds with the dilaton field turned off [2]. He also classified all the ten-dimensional parallelizable NS-NS field configurations which do not necessarily satisfy the equations of motion of type II super-gravity but with a closed torsion in the same paper [2]. This is the classification of the ten-dimensional parallelized Lorentzian Lie groups with a bi-invariant metric. In this short note, we will turn on the dilaton field to classify all the dilatonic par-allelizable NS-NS backgrounds in type II supergravity. Our strategy is simple: we will find the solution of the dilaton field to the equations of motion which is facilitated by the parallelizability of the geometry. It turns out that all the manifolds in the classification