When is Riemannian, there exists a natural embedding of the Clifford algebra into
This embedding depends on a sign (for “embedding”) and has the property that . Thus, it
embeds into . When is odd, the dimension of the minimal spinor
representation in is double the one in . In this case, the Clifford-valued volume form in both
manifolds is mapped as follows
Thus, spinors in will be mapped to spinors of a definite chirality in .
Plugging this embedding into the expression for , one sees that a -parallel spinor in the
cone, restricts to to a geometric Killing spinor obeying
This is the defining equation for a geometric Killing spinor. Furthermore,
- if is even: there exists a one-to-one correspondence between parallel spinors in
and geometric Killing spinors in ; and
- if is odd: there exists a one-to-one correspondence between parallel spinors in of
eigenvalues and geometric Killing spinors in .