Reflections and spinors on manifolds
  This paper reviews some recent work on (s)pin structures and the Dirac operator on hypersurfaces (in particular, on spheres), on real projective spaces and quadrics. Two approaches to spinor fields on manifolds are compared. The action of space and time reflections on spinors is discussed, also for two-component (chiral) spinors. 
Andrzej Trautman