= Tensorial and directional second covariant derivative commutators
For a <vector field> $V$, the tensorial second derivative is $\nabla^2V(X,Y)=\nabla_X\nabla_YV-\nabla_{\nabla_XY}V$. With $R(X,Y)V=\nabla_X\nabla_YV-\nabla_Y\nabla_XV-\nabla_{[X,Y]}V$ and geometric <torsion tensor> $\mathcal T$, antisymmetrization gives
$$
\nabla^2V(X,Y)-\nabla^2V(Y,X)=R(X,Y)V-\nabla_{\mathcal T(X,Y)}V.
$$
This follows by subtracting the two definitions and using $\nabla_XY-\nabla_YX=[X,Y]+\mathcal T(X,Y)$. By contrast, the directional operator commutator subtracts no derivative-index connection and obeys $[\nabla_X,\nabla_Y]V=R(X,Y)V+\nabla_{[X,Y]}V$. On scalars the directional commutator is $[X,Y]f$, while the tensorial Hessian commutator is $-\mathcal T(X,Y)f$. Distinguishing these objects prevents an apparent contradiction when a scalar Hessian has an antisymmetric part despite commuting coordinate vector fields.
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