Principles of Advanced Mathematical Physics pp 114-128 | Cite as

# Covering Manifolds

The 2-to-1 mapping *ψ* of SU(2) onto SO(3) found in Section 19.7 is more than merely a group homomorphism. It is also a mapping of the manifold of SU(2) onto the manifold of SO(3) of the kind known as a covering. It is locally a homeomorphism, in the sense that if *P* is any point on the manifold of SU(2), and *Q* is its image in the manifold of SO(3), then *P* has a neighborhood that is mapped homeomorphically by *ψ* onto a neighorhood of *Q*. Furthermore, if *Q* is *any* point of the second manifold, then there are always two points *P* in the first manifold, each of which has such a neighborhood. *ψ* is a *two-sheeted covering* of SO(3) by SU(2).

## Keywords

Local homeomorphism Projection*p*-sheeted covering good neighborhood principles of lifting universal covering manifold construction of mathematical models manifolds covered by a given manifold

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