Weyl's own 1918 gauge program shows that projective and conformal structures together determine the affine structure, so projective structure alone underdetermines zero acceleration.
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A geometric structure on a manifold that determines only an equivalence class of conformally equivalent symmetric linear connections, not a unique affine connection.
projective structure(Contrasted with the infinite-parameter microsymmetry group present when only a differentiable structure exists.)
A post-differentiable topological geometric field on spacetime whose preservation reduces the microsymmetry group at any event to a 20-parameter Lie group.
underdetermines(logic and language)
Doesn't fully decide or pin down; leaves open multiple possible interpretations because there isn't enough information visible to choose just one.