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Here is a result of Anghel (see BaerExtrinsicBoundsDirac) that is useful in producing low eigenvalues of Dirac on a hypersurface of $S^4$.  Let $\Sigma$ be an $n$-dimensional closed oriented hypersurface in $S^{n+1}$.  Let $\Sigma$ carry the induced spin structure with mean curvature $H$.  Then the intrinsic Dirac operator $D_\Sigma$ has at least $2^{[n/2]}$ eigenvalues $\lambda$ satisfying
$\lambda^2 \le \frac{n^2}{4} + \frac{n^2}{4 vol(\Sigma)}\int H^2$
In the S4 picture, this shows that if the physical universe has bounded mean curvature, we can expect it to have some matter-energy.  We identify matter-energy in $\Sigma$ to eigenspinors of the Dirac operator on $\Sigma$ for particular issues but generally we identify matter-energy with eigenspinors of $S^4(1/h)$.  I am still trying to understand the relation of the physics to both these identifications.
For a hypersurface $\Sigma \i M$ the relation between the spin structures are well-known.  The restriction of the spin bundle $SM$ of $M$ to $\Sigma$ can be identified, when the dimension of $\Sigma$ is odd to $SM |_\Sigma = S\Sigma\oplus S\Sigma$.  There are two Dirac operators on $S\Sigma$, the intrinsic one as well as the extrinsic Dirac operator which differ by a second fundamental form term.  The latter is needed to work with spinors in the ambient manifold.  Real Killing spinors exist on spheres which makes it worthwhile to study them on the ambient space.