For each Riemannian metric g on a manifold there exists a unique torsion-free connection preserving g, the celebrated Levi-Civita connection. Conversely one can try to characterise the connections preserving a metric. A concrete characterisation easily applicable to examples was given by L. Eisenhart and O. Veblen in 1922. One can also study the problem of characterising the connections which are only projectively equivalent to a metric connection (i.e. share the same unparametrised geodesics with a metric connection). The latter problem, albeit first studied by R. Liouville in 1889, was solved only recently.
In this talk, after discussing the aforementioned results, I will explain why locally on a surface every connection is projectively equivalent to a connection preserving a conformal structure (a so-called Weyl connection).
Surprisingly, the relevant PDE corresponds to the Cauchy-Riemann equations.
This allows to classify the Weyl connections on the 2-sphere whose geodesics are the great circles using techniques from algebraic geometry.