A-geometrical approach to Topological Insulators with defects

The study of the propagation of electrons with a varying spinor orientability is performed using the coordinate transformation method. Topological Insulators are characterized by an odd number of changes of the orientability in the Brillouin zone. For defects the change in orientability takes place for closed orbits in real space. Both cases are characterized by nontrivial spin connections. Using this method , we derive the form of the spin connections for topological defects in three dimensional Topological Insulators. On the surface of a Topological Insulator, the presence an edge dislocation gives rise to a spin connection controlled by torsion. We find that electrons propagate along two dimensional regions and confined circular contours. We compute for the edge dislocations the tunneling density of states. The edge dislocations violates parity symmetry resulting in a current measured by the in-plane component of the spin on the surface.