Spiro Karigiannis (University of Waterloo)
Decomposition of Riemann curvature in 4 dimensions (and beyond?)
This is a continuation of my earlier talk from May. We showed that the Riemann curvature tensor of a Riemannian metric decomposes into three orthogonal components: the scalar curvature, the traceless Ricci curvature tensor, and the Weyl curvature tensor. We will see that in dimension 4 we can say more. The Weyl curvature decomposes into two pieces, and we will explicitly determine the decomposition the curvature operator (as a self-adjoint operator on 2-forms). As an application, we will discuss the Singer-Thorpe Theorem characterizing 4-dimensional Einstein metrics in terms of this decomposition. If time permits, we will briefly discuss a generalization of these ideas to G2-geometry in seven dimensions.
MC 5417