= Skew-adjoint difference criterion for metric connections
{title2=$g(A(X,Y),Z)+g(Y,A(X,Z))=0$}
Let $\nabla$ be a <metric connection> for a <Riemannian metric> $g$, and let $\nabla'$ be another <affine connection> on the same <tangent bundle>. The <difference of affine connections is a tensor> $A=\nabla-\nabla'$. Expanding the derivative of the <metric tensor> gives
$$
(\nabla'_Xg)(Y,Z)=g(A(X,Y),Z)+g(Y,A(X,Z)).
$$
Thus $\nabla'$ is a <metric connection> precisely when each endomorphism $A(X,\cdot)$ is skew-adjoint. This imposes no antisymmetry in $X,Y$; that separate condition characterizes agreement of parametrized <geodesics>. If both conditions hold, $g(A(X,Y),Z)$ is alternating in all three arguments.
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