  The method of the space dependent basis is applied to study electronic
spinors in a crystal. The crystal in the momentum space is described by the
Brillouine zone which might contains obstructions or degeneracies for which
requires different gauges for different regions. The electronic bands are
classified according to their topology. The connection and curvature determines
the physical properties which are clasified according to the topological
invariants. We apply this method to the Topological Insulators, Topological
Superconductors, Persistent Currents in coupled rings and photoemission for a
curved crystal-face boundary
