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arXiv · 1410.8501

Geodesic rigidity of conformal connections on surfaces

Thomas Mettler
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Abstract

We show that a conformal connection on a closed oriented surface $Σ$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $Σ$ determine the metric up to constant rescaling. It is also shown that every conformal connection on the $2$-sphere lies in a complex $5$-manifold of conformal connections, all of which share the same unparametrised geodesics.

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Thomas Mettler. 2015-05-09. Geodesic rigidity of conformal connections on surfaces. https://doi.org/10.1007/s00209-015-1489-5

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