A note on the conformal geometry of the 2-sphere

Abstract

We discuss the geometry of the two-dimensional sphere. Our perspective focuses on the set of conformal factors that rescale a given metric to a round metric. We find that there exists a unique symmetric and trace-free tensor on the sphere which complements the Ricci tensor to a divergence-free symmetric tensor. Using this tensor, we derive a linear second-order PDE whose solution space carries a natural Minkowski structure. We show that there are in fact infinitely many such spaces, all related by isometries which can be thought of as being parametrised by ``4-vectors''. Finally, we show that every 2-sphere can be isometrically embedded in a Minkowski space in a canonical way.