Killing spinors are Killing vector fields in Riemannian supergeometry

A supermanifold M is canonically associated to any pseudo-Riemannian spin manifold (M-0, g(0)), Extending the metric g(0) to a field g of bilinear forms g(p) on TpM, p is an element of M-0, the pseudo-Riemannian supergeometry of(M, g) is formulated as G-structure on M, where G is a supergroup with even part G(0) congruent to Spin(k, l); (k, l) the signature of (M-0, g(0)). 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 X-s on M. Our main result is that X-s is a Killing vector field on (M, g) if and only if s is a twister spinor. In particular, any Killing spinor s defines a Killing vector field X-s.