B.1 (M, g) Riemannian

When (M, g) is Riemannian, there exists a natural embedding of the Clifford algebra Cℓ(d,0) into even C ℓ(d + 1,0):
even ^ ^ Cℓ(d,0 ) `→ C ℓ(d + 1,0) where Γ i ↦→ 𝜀Γ iΓ r. (562 )
This embedding depends on a sign 𝜀 (for “embedding”) and has the property that Γ ij ↦→ ^Γ ij. Thus, it embeds spin(d,0) into spin(d + 1,0). When d is odd, the dimension of the minimal spinor representation in M^ is double the one in M. In this case, the Clifford-valued volume form ν in both manifolds is mapped as follows
νd,0 ↦→ 𝜀νd+1,0. (563 )
Thus, spinors in M will be mapped to spinors of a definite chirality in ^M.

Plugging this embedding into the expression for ^∇, one sees that a ^∇-parallel spinor ψ^ in the cone, restricts to (M, g ) to a geometric Killing spinor ^ ψ = ψ|r=R obeying

𝜀-- ∇X ψ = − 2R X ⋅ ψ. (564 )
This is the defining equation for a geometric Killing spinor. Furthermore,
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