Symmetric Hessian of a scalar field

For a torsion-free connection, $U_{\mu }={\partial }_{\mu }\varphi$ satisfies ${\nabla }_{\alpha }{U}_{\beta }={\nabla }_{\beta }{U}_{\alpha }$.

Use the sequential convention ${\phi }_{;lmn}={\nabla }_n{\nabla }_m{\nabla }_l\phi$ and define $R$ by $X^a{}_{;mn}-X^a{}_{;nm}=R^a{}_{bmn}{X}^b$. The symmetric Hessian of a scalar field gives symmetry in $l,m$, but the curvature commutator on a covariant tensor gives ${\phi }_{;l[mn]}=-R^p{}_{lmn}{\phi }_{;p}/2$. Averaging the three distinct permutations yields ${\phi }_{;(lm)n}={\phi }_{;(lmn)}-(R^p{}_{lmn}+R^p{}_{mln}){\phi }_{;p}/3$. Thus a scalar's third covariant derivative is not generally totally symmetric.