= Antisymmetric covariant Hessian measures torsion
{title2=$\nabla^2f(X,Y)-\nabla^2f(Y,X)=-\mathcal T(X,Y)f$}
For an <affine connection>, define geometric <torsion tensor> by $\mathcal T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]$. The covariant Hessian of a <scalar field> is $\nabla^2f(X,Y)=X(Yf)-(\nabla_XY)f$. Antisymmetrizing gives $[X,Y]f-(\nabla_XY-\nabla_YX)f=-\mathcal T(X,Y)f$. 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.
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