1. Geodesic rigidity of conformal connections on surfaces
- Author
-
Thomas Mettler
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Geodesic ,General Mathematics ,Conformal map ,Twistor space ,Projective structures ,Mathematics - Algebraic Geometry ,symbols.namesake ,Geodesic rigidity ,Euler characteristic ,Conformal connections ,FOS: Mathematics ,Primary 53A20 ,Secondary 53C24 ,53C28 ,Algebraic Geometry (math.AG) ,Mathematics ,Quantitative Biology::Biomolecules ,Surface (topology) ,Manifold ,Differential Geometry (math.DG) ,Metric (mathematics) ,symbols ,Conformal connection ,Mathematics::Differential Geometry - Abstract
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., Comment: 16 pages, exposition improved, references added
- Published
- 2021