B.2 (M, g) of signature (1, d − 1 )

When (M, g) is Lorentzian, there also exists a natural embedding of the Clifford algebra C ā„“(1,d − 1) into C ā„“(2,d − 1)even depending on a sign šœ€:
even ^ ^ Cā„“(1,d − 1) `→ C ā„“(2, d − 1 ) where Γ i ā†¦→ šœ€Γ iΓ r . (565 )
This embedding induces an embedding spin(d,0) `→ spin (d + 1,0), Γ ij ā†¦→ ^Γ ij. Moreover if d is odd, one has
ν1,d− 1 ā†¦→ šœ€ν2,d−1. (566 )
Plugging this into the expression for ^∇, we see that a ^∇-parallel spinor ^ψ in the cone, restricts to (M, g ) to a geometric Killing spinor ^ ψ = ψ|r=R obeying
∇ ψ = − -šœ€-X ⋅ ψ . (567 ) X 2R
Furthermore,
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