Antisymmetric covariant Hessian measures torsion (${\nabla }^{2}f(X,Y)-{\nabla }^{2}f(Y,X)=-\mathcal{T}(X,Y)f$)

For an affine connection, define geometric torsion tensor by $\mathcal{T}(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$. The covariant Hessian of a scalar field is ${\nabla }^{2}f(X,Y)=X(Yf)-({\nabla }_{X}Y)f$. Antisymmetrizing gives $[X,Y]f-({\nabla }_{X}Y-{\nabla }_{Y}X)f=-\mathcal{T}(X,Y)f$. Thus all scalar Hessians are symmetric exactly when the connection is torsion-free. In derivative-first coordinates, ${\nabla }_{\mu }{\nabla }_{\nu }f={\partial }_{\mu }{\partial }_{\nu }f-{\Gamma }_{\mu \nu }{}^{\lambda }{\partial }_{\lambda }f$, which makes the sign explicit. Differentials of local coordinate functions span the cotangent space, so vanishing of this antisymmetric part for every $f$ forces the torsion itself to vanish.