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this is my first time posting so please point out if I am doing it wrong.

So as the title says my question is, What is the relation between "projective geometry" and "projective geometry of paths"?

Projective geometry, for all I know, is more frequently researched in mathematics. It could be found for example in Wikipedia as https://en.wikipedia.org/wiki/Projective_geometry
and what I read it is a purely mathematical concept with some application to physics, such as quantum mechanics.

https://math.stackexchange.com/questions/104729/example-for-projective-geometry-used-in-quantum-mechanics

"Projective geometry of paths" is a special set of affine-connections that relates to one another as \begin{align} \Gamma^\lambda_{~\mu\nu}\to\Gamma^\lambda_{~\mu\nu}+A_\mu \delta^\lambda_\nu +A_\nu \delta^\lambda_\mu \end{align} which results in the same geodesic equation with respect to the redefinition of the affine parameter. It was considered by Weyl, Schouten, Thomas, and many others in the context of General Relativity and it's extensions.

"Ricci Calculus" by Schouten (Springer,1954) https://projecteuclid.org/euclid.bams/1183486553

https://www.jstor.org/stable/1968625

Since the latter has "projective geometry" in it, and furthermore some researchers actually use "projective geometry" for describing the latter, I expected that there is some relation between them. However, the former has no relation with the geodesics it seems, while the geodesics are crucial to the latter.

Can anyone tell me what I am missing?

Maybe related questions

Is projective geometry important for general relativity?

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