Luca Seemungal (University of Leeds) – Curvature and Topology

That the angles of a plane triangle sum to \(\pi\) is a consequence of the parallel postulate. Hiding, therefore, in this formula from primary school is a curvature assumption. It was only a couple of millennia later that mathematicians decided to start considering curved geometries, without the parallel postulate, yielding for example formulae for the sum of the angles of spherical triangles. The conclusion of all this is the shocking Gauss—Bonnet theorem, which states that the curvature (a local property) of a surface determines its topology (a global property).