= Projective equivalence of affine connections
{title2=$\bar\Gamma^a{}_{bc}-\Gamma^a{}_{bc}=\delta^a_bV_c+\delta^a_cV_b$}
= Projectively equivalent connections
{synonym}
= Projectively equivalent
{synonym}
Two <torsion-free connections> are <projectively equivalent> if they have the same unparametrized <geodesics>. Their <difference of affine connections is a tensor> $S$. A change of parameter adds only tangential acceleration, so equality of curves requires $S(v,v)$ to be parallel to $v$ for every tangent vector. The <symmetric bilinear diagonal-parallel lemma> gives exactly the displayed condition. Conversely the <geodesic reparametrization under projective equivalence> absorbs this tangential term. Agreement with the same <affine parameters> instead forces $S=0$.
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