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Dirac operators are exhaustively well-understoond and well-studied on compact riemannian manifolds.  One can construct Dirac operators from connections on principal bundles by introducing a Clifford bundle and defining Dirac operators using Clifford multiplication by the basis vectors.  An interesting feature of a 4-sphere scaled by $1/h$ is that the spectrum of the Dirac operator are integer multiples of $h$.  This fact seems quite significant for theory of physics for a 4-sphere universe but I have not yet found the proper significance.