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RESTRICTION OF REAL KILLING SPINORS TO HYPERSURFACES

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: $\Sigma N|_M = \Sigma M$ when the dimension of $M$ is even; otherwise one takes the splitting $\Sigma N = \Sigma^+ N \oplus \Sigma^- N$ based on the $+/-$-eigenvalues of the action of the complex volume form $i^{[n+1/2]}\omega$ and finds $\Sigma M = \Sigma^+ N$.  One can then check that spinor fields $\psi$ on $N$ restrict to $M$ via $\psi\rightarrow \psi^*=\nu \cdot \psi$.  The spinorial Gauss formula is

$(\nabla^N \psi)^* = \nabla \psi + \frac{1}{2} h(X)\cdot \psi$

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

$T^{\psi}(X,Y) = Re\langle X\cdot\nabla_Y \psi + Y\cdot \nabla_X \psi, \psi / |\psi|^2 \rangle$

We want to show that when

$2 T^{\psi} = h + 6 \lambda$

For a tangential orthonormal frame $e_1, e_2, e_3$ of the three-manifold, we apply the spinorial Gauss formula with $X=e_i$, Clifford multiply with $e_j$ and take inner product with $\psi$.  We obtain

$Re\langle e_j \nabla^N_{e_i}\psi+ e_i \nabla^N_{e_j}\psi, \psi \rangle = Re\langle e_j \nabla_{e_i} \psi + e_i \nabla_{e_j} \psi, \psi\rangle + \frac{1}{2} ( e_j \cdot h(e_i) + e_i \cdot h(e_j))\cdot \psi, \psi\rangle$

On the first part use the real Killing spinor condition and on the right use the symmetry of $h(\cdot)$ and on both sides use the Clifford relation $e_i e_j + e_j e_i = \delta_{ij}$ and sum over $i,j=1,2,3$.  The result is

$-3\lambda |\psi|^2 = T^{\psi} |\psi|^2 - \frac{1}{2} \sum_{i,j} h_{ij} |\psi|^2$

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)