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Studying the scalar curvature of a hypersurface $M$ of $S^4$ is one way of trying to show nonexistence of harmonic spinors (positive scalar curvature would do).  Another approach is to assume that a nontrivial harmonic spinor exists on $M$ and try to extend it to a harmonic spinor on $S^4$ by solving an appropriate boundary value problem.  If there is a canonical procedure for doing this, then we can use the positive scalar curvature of $S^4$ to show that the extension must be zero.  The physical problem of interest is the existence of massless fermions.