= Third covariant derivatives of a scalar
Use the sequential convention $\varphi_{;lmn}=\nabla_n\nabla_m\nabla_l\varphi$ 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 $\varphi_{;l[mn]}=-R^p{}_{lmn}\varphi_{;p}/2$. Averaging the three distinct permutations yields $\varphi_{;(lm)n}=\varphi_{;(lmn)}-(R^p{}_{lmn}+R^p{}_{mln})\varphi_{;p}/3$. Thus a scalar's third covariant derivative is not generally totally symmetric.
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