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  "body": "\n## Summary\n\nA projective connection is the structure a manifold carries when it knows which curves are its straight lines.\nIt does not know how these lines are parametrized.\nIn Cartan's language it is a [[Cartan connection|QPR7DCHXPG]] modelled on real projective space, viewed as a homogeneous space of the projective linear group.\nClassically it is described by a class of torsion-free affine connections that have the same geodesics up to reparametrization.\nProjective connections appear in projective differential geometry and in the study of second-order ordinary differential equations.\n\n## Explanations\n\n### L1 human\n\nIn projective geometry there are lines but no distances and no preferred speed along a line.\nA projective connection gives a curved space the same kind of structure: a family of curves that play the role of straight lines.\nTwo different ways of measuring acceleration can produce exactly the same paths with different speeds; they define the same projective connection.\nThe flat example is projective space itself, or ordinary space with its usual straight lines.\n",
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