Bochner formulae for orthogonal G-structures on compact manifolds

A general technique is introduced for deriving Bochner type formulae on a compact riemannian manifold, relating its curvature tensor with the intrinsic torsion of a compatible (orthogonal) G-structure. The technique is illustrated for the groups G=Un,SUn,G2 and Spin7, with various applications of the derived formulae in these cases.