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In LawnRothIsometricImmersionS4Spinors Lawn and Roth gave a sufficient condition for a three manifold to be isometrically immersed into a scaled 4-sphere. Their results are slightly more general but we are interested only in the case of isometric immersion into a 4-sphere. The condition is that if there exist two spinor fields $\phi_1$ and $\phi_2$ that satisfy the Dirac equations
$D\phi_1 = (3/2 H + 3/2 \kappa)\phi_1$ and $D\phi_2 = -(3/2 H + 3/2 \kappa)\phi_2$
where $H$ can be a function on the manifold rather than simply a constant, then the three-manifold can be isometrically embedded in a 4-sphere of curvature $\kappa^2$ in such a way that $H$ will appear as the mean curvature of the embedding.
This theorem can be used to show that the physical universe can be isometrically immersed in S4(1/h) is there exists in the physical universe a massive fermion and it antiparticle. In general it is useful to analyze the static eigenspinors noting that the time-dependent Dirac equation will be satisfied by multiplying these eigenspinors by $e^{i \omega t}$ just as is usually done to solve wave equations using Laplace eigenfunctions. So if we assume the physical universe is a compact three-dimensional manifold $M$ and that a massive fermion and its antiparticle exists, then we have eigenspinors with eigenvalue $3/2 m$, the mass and its antiparticle of opposite signed mass. Since energy is quantized in units of $h$ we can assume $3/2(m - h) \ge 0$ and we see that we have fulfilled the Lawn-Roth necessary conditions with $H=m-h$ and therefore $M$ can be isometrically immersed in S4(1/h) with mean curvature $H$.