# Projectively flat Weyl connection on closed higher genus surface

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 $$g\geqslant 2$$?

• Thomas Mettler will know and probably only a few other people. – Ben McKay Nov 13 '18 at 19:07
• 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. – Ben McKay Nov 13 '18 at 19:11
• 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. – Ben McKay Nov 13 '18 at 19:17
• Thank you Ben. I will have a look at Agaoka's paper. – user113646 Nov 13 '18 at 19:35