 Tensorial and directional second covariant derivative commutators

For a vector field $V$, the tensorial second derivative is ${\nabla }^{2}V(X,Y)={\nabla }_X{\nabla }_{Y}V-{\nabla }_{{\nabla }_{X}Y}V$. With $R(X,Y)V={\nabla }_X{\nabla }_{Y}V-{\nabla }_Y{\nabla }_{X}V-{\nabla }_{[X,Y]}V$ and geometric torsion tensor $\mathcal{T}$, antisymmetrization givesThis follows by subtracting the two definitions and using ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]+\mathcal{T}(X,Y)$. 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.