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Uniqueness of the stereographic embedding. (English) Zbl 1340.53012

The author studies the question of the conformal compactification of \(\mathbb{R}^n\), \(n \geq 3\). It is not suprising that the only such compactification is \(S^n\) with the round metric. It was proved in [C. Frances, “Rigidity at the boundary for conformal structures and other Cartan geometries”, Preprint, arxiv:0806.1008]. The author’s aim is to provide a much more elementary proof based on so-called conformal geodesics. Their characterization is well known but significantly more complicated than for Riemannian geodesics. After a nice summary of properties of conformal geodesics, and using the notion of so-called highly accessible points at the boundary, author provides a very elementary (and elegant) proof.

MSC:

53A30 Conformal differential geometry (MSC2010)
53C22 Geodesics in global differential geometry
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