Tangent Bundle on Three Dimension Small Cover

Small covers arising from 3-dimensional simple polytopes. The geometry of the tangent bundle of a three dimension small cover is an old topic. The theory is of great importance in mathematics and physics. It is an interesting question to understand whether tangential bundles exist on the 3-dimensional small cover. In this paper a three dimension small cover on the tangent bundle is defined by using a spin structure on the compact and oriented smooth three manifold. The SO(3) is diffeornorphic to the unit tangent bundle of the two-sphere. Since the SO(3) is diffeornorphic to RP3. The line bundle over RP3 is just trivial bundle. We use fiber bundle pullbacks measures prove that 3-dimensional small cover has a trivial tangent bundle. To the knowledge of the author of this paper, we only considered the special case on small cover and effectively connected Clifford algebras, spin groups, and small cover. Note that the general case is much more complicated. Some results in the general case were not proved.