15 oct 2014 -- 14:30
Aula di Consiglio, Dip. di Matematica dell'Università "La Sapienza", Roma
Abstract.
A definite connection is an SO(3)-connection over a 4-manifold, whose curvature is non-zero on every tangent 2-plane. Such geometric objects were first considered by Allan Weinstein and were called fat bundles. Given such a connection, the associated S2-bundle is naturally a symplectic manifold. Surprisingly the symplectic manifold is either a symplectic Fano or a symplectic Calabi-Yau. Important examples of such connections come from differential geometry. Namely, the Levi-Cevita connection on the twistor bundle of a Riemannian 4-manifold with sufficiently pinched curvature is definite. In particular we have examples such as the unit S4 that give rise to positive definite connections and examples such as hyperbolic 4-manifolds that give rise to negative definite connections. In this talk I will discuss the following sphere-type theorem: the only four manifolds that admit an S1-invariant definite connection are S4 and CP2. This work is joined with Joel Fine.