On the Spinor Structure of the Homogeneous and Isotropic Universe in Closed Model

X iv :0 70 8. 11 71 v1 [ m at h. D G ] 8 A ug 2 00 7 ON THE SPINOR STRUCTURE OF THE HOMOGENEOUS AND ISOTROPIC UNIVERSE IN CLOSED MODEL. R. A. Sharipov Abstract. The closed homogeneous and isotropic universe is considered. The bundles of Weyl and Dirac spinors for this universe are explicitly described. Some explicit formulas for the basic fields and for the connection components in stereographic and in spherical coordinates are presented. 1. Stereographic projections of the sphere S. It is known that the closed homogeneous and isotropic universe is described by a manifold diffeomorphic to the Cartesian product of the three-dimensional sphere S by a straight line: M = R × S (see § 111 and § 112 in [1]). The sphere S is a manifold that can be covered with two local charts. We choose stereographic projections from two diametrically opposite points (we call them North and South poles) to their equatorial hyperplane. The sphere S is naturally presented as a threedimensional hypersurface in the four-dimensional space R with the standard Euclidean metric. Let z be a point on such a sphere with the radius R. Then z/R is a point on the unit sphere. Let x and y be the stereographic projections of z/R. Then the points x and y lie on some ray in the equatorial hyperplane coming out from the center of the sphere. It is rather easy to derive the following formulas relating their coordinates: