A Weyl connection on a smooth $n$-manifold $M$ is a torsion-free connection $\nabla$ on its tangent bundle that preserves some conformal structure $[g]$ on $M$. By this I mean that its parallel transport maps are angle preserving with respect to $[g]$.

A torsion-free connection $\nabla$ on $TM$ is called projectively flat if for every point $p \in M$ there exists a neighbourhood $U_p$ and a diffeomorphism $\psi : U_p \to V \subset \mathbb{R}^n$ so that the geodesics of $\nabla$ contained in $U_p$ are mapped onto (segments of) straight lines. It is a classical result that the Levi-Civita connection of a Riemannian metric is projectively flat if and only if the metric has constant sectional curvature.

Is there an example of a non-Riemannian projectivley flat Weyl connection on a closed oriented surface of genus $g\geqslant 2$?

  • $\begingroup$ Thomas Mettler will know and probably only a few other people. $\endgroup$ – Ben McKay Nov 13 '18 at 19:07
  • $\begingroup$ Agaoka's paper Geometric invariants associated with flat projective structures is worth looking at in this regard, but I think his examples are all in dimension 3 and higher. $\endgroup$ – Ben McKay Nov 13 '18 at 19:11
  • $\begingroup$ By Mettler's result, every projectively flat torsion free connection is projectively equivalent to a Weyl connection, so you have to construct a projectively flat torsion free connection whose geodesics are not those of any Riemannian metric, I think. $\endgroup$ – Ben McKay Nov 13 '18 at 19:17
  • $\begingroup$ Thank you Ben. I will have a look at Agaoka's paper. $\endgroup$ – user113646 Nov 13 '18 at 19:35

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