Recall that a collineation $\sigma$ on a real affine/projective space is a self-bijection of the space which preserves lines (as sets of points) of the space. Considering that geodesics are a natural extension of "lines" on Riemann manifolds of dimension $\ge 2$, a "cogeodesation" $\tau$ on a complete Riemann manifold can be similarly defined to be a bijection of the manifold which preserves geodesics (as sets of points).
Question:
$1$: The weaker line-keeping property of $\sigma$ will lead to its stronger linear-mapping nature. Does $\tau$ have a similar property, that we can deduce its smoothness (thus a projective automorphism) from its geodesic-preserving property?
$2$: A seemingly weaker but equivalent definition for a collineation $\sigma$ is that $\sigma$ maps any $3$ collinear points to $3$ still collinear points. Do we get the same set of cogeodesations (no need to be smooth if the answer to question $1$ is negative) if we loose the definition to 'mapping any $3$ cogeodesic points to $3$ still cogeodesic points'?