 Projective equivalence of affine connections

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 $S=0$.