20 feb 2025 -- 14:00
room 2109 (U5 building), UniMiB
Abstract.
In this talk, we will briefly present some unique features of four-dimensional Riemannian Geometry and their connections with twistor theory: indeed, it is well known that four-dimensional Riemannian manifolds carry many peculiar properties, which give rise to the existence of unique special metrics (e.g. half conformally flat metrics). In their study of self-dual solutions of Yang-Mills equations, Atiyah, Hitchin and Singer adapted the celebrated Penrose’s construction of twistor spaces to the Riemannian context, showing that a Riemannian four-manifold is half conformally flat if and only if its twistor space is a complex manifold: this paved the way for the study of many other characterizations of curvature properties for Riemannian four-manifolds. After giving an overview of the basic properties of twistor spaces in the four-dimensional case, we present some new rigidity results for Riemannian four-manifolds whose twistor spaces satisfy specific vanishing curvature conditions. This is based on joint works with Giovanni Catino and Paolo Mastrolia.