Some Spinor-Curvature Identities
  We describe a class of spinor-curvature identities which exist for Riemannian or Riemann-Cartan geometries. Each identity relates an expression quadratic in the covariant derivative of a spinor field with an expression linear in the curvature plus an exact differential. Certain special cases in 3 and 4 dimensions which have been or could be used in applications to General Relativity are noted. 
James M. Nester, Roh Suan Tung and Vadim V. Zhytnikov