Jean Pierre Bourguignon

The many facets of curvature Recalling several moments where General Relativity influenced Mathematics

The Ricci curvature was introduced by Grigorio Ricci-Curbastro for purely geometric reasons, and its appearance in the field equations of General Relativity changed its status in Geometry.

All along the XXth century, there has been instances where new developments in relativistic field theory changed the point of view mathematicians had on some geometric quantities or problems. We will recall some of these moments, typically the early days of the Ricci curvature, Hermann Weyl's interest in conformal geometry, the introduction of the Dirac operator and the role of spinor fields, generalized Gauß-Bonnet theorems, and the ADM formalism leading to super-space and the Ricci flow. Almost each time the mathematical techniques needed to take advantage of the new point of view had to be considerably expanded.