Topologically nontrivial solution in Einstein-Dirac gravity on the Hopf bundle

The topologically nontrivial solution in Einstein-Dirac gravity with cosmological constant is obtained. The spacetime has the Hopf bundle as a spatial section. It is shown that the Hopf invariant is related to the spinor current density. Two Dirac spinors are used for obtaining a diagonal energy-momentum tensor. The solutions for the nongravitating Dirac equation on the background of Lorentzian spacetime with the Hopf bundle as a spatial section are also obtained. Nongravitating solutions of the Dirac equation are defined by two quantum half-integer numbers $m,n$.