= Torsion-free connections are determined by their parametrized geodesics
The difference of two <torsion-free connections> is symmetric, because subtracting their zero-torsion identities gives $\Delta(X,Y)=\Delta(Y,X)$. Agreement of parametrized <geodesics> makes this difference skew-symmetric as well, by <parametrized geodesics determine the symmetric part of an affine connection>. Over the real numbers it must therefore vanish. Without the torsion-free condition, an arbitrary skew-symmetric difference changes torsion without changing the parametrized <geodesics>.
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