Geodesic rigidity of conformal connections on surfaces
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We show that a conformal connection on a closed oriented surface \(\Sigma \) 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 \(\Sigma \) 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.
KeywordsProjective structures Conformal connections Geodesic rigidity Twistor space
Mathematics Subject ClassificationPrimary 53A20 Secondary 53C24 53C28
This paper would not have come into existence without several very helpful discussions with Nigel Hitchin. I would like to warmly thank him here. I also wish to thank Vladimir Matveev for references and the anonymous referee for her/his careful reading and useful suggestions.
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