Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 59 / 1 / c / Solution

For a torsion-free connection, the covariant Hessian of a scalar is symmetric, so $[{\nabla }_a,{\nabla }_b]f=0$. Moreover, a commutator of two covariant derivatives acts as a derivation on tensor products. Applying the Leibniz rule twice, the terms containing one derivative on each factor cancel between the two orders:Projecting with the parallel symplectic form preserves this identity for the chiral spinor curvature operator. Since its scalar projection obeys ${\Delta }_{AB}f=0$, we obtainThus the chiral spinor curvature operator is algebraic over scalar functions even though its displayed expression uses second derivatives. If torsion tensor is allowed, $[{\nabla }_a,{\nabla }_b]f=-T^c{}_{ab}{\partial }_{c}f$ need not vanish, explaining why metric compatibility alone would not prove the claim.