Killing spinors are killing vector fields in Riemannian supergeometry

A supermanifold M is canonically associated to any pseudo-Riemannian spin manifold (M0, g0). Extending the metric g0 to a field g of bilinear forms g(p) on TpM, p ϵ M0, the pseudo-Riemannian supergeometry of (M, g) is formulated as G-structure on M, where G is a supergroup with even part G0 ≊ Spin(k, l); (k, l) the signature of (M0, go). Killing vector fields on (M, g) are, by definition, infinitesimal automorphisms of this G-structure. For every spinor field s there exists a corresponding odd vector field Xs on M. Our main result is that Xs is a Killing vector field on (M, g) if and only if s is a twistor spinor. In particular, any Killing spinor s defines a Killing vector field Xs.