Singlet fermions on curved extra dimension tori

A model with two curved compact extra dimensions is introduced. The model is based on a four-brane immersed in a six dimensional space, where the extra dimensions are compact but not flat. They have topology of torus. The form of the metric in the empty bulk is studied and the gauge singlet fermion structure showed to be very simple. It contain only one massless low energy mode which couples to brane matter, while the massive modes are not related to the volume of torus while the Planck mass is related to the volume of extra dimensions. In the model "our" brane is situated on a nearly singular line of the torus.