A general fact about spin structures on hypersurfaces $M\i N^{n+1}$ is that the intrinsic spin structure is related to the restriction of the spin structure on the ambient manifold as follows: when the dimension of is even; otherwise one takes the splitting based on the -eigenvalues of the action of the complex volume form and finds . One can then check that spinor fields on restrict to via . The spinorial Gauss formula is

where is the second fundamental form. Real spinor fields with parameter on the ambient manifold satisfy . These exist with (here we abuse notation and mean a Planck constant for ) for Slatex S^4(1/h)$. Let the energy-momentum tensor for a spinor field be defined as

We want to show that when

For a tangential orthonormal frame of the three-manifold, we apply the spinorial Gauss formula with , Clifford multiply with and take inner product with . We obtain

On the first part use the real Killing spinor condition and on the right use the symmetry of and on both sides use the Clifford relation and sum over . The result is

The importance of this computation is that it tells us that the energy-momentum tensor defined in terms of the spinor field can be expressed as a second fundamental form with the lambda term. The lambda term will directly correspond to a cosmological constant in Einstein’s gravitational field equations for example in the approach of Friedrich and Kim on solving Dirac-Einstein equations (MorelEnergyMomentumSecondFundamentalForm2003EinsteinDiracRiemannianSpin)

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