Spiro Karigiannis, University of Waterloo
Decomposition of the Riemann Curvature Tensor
The Riemann curvature tensor R of a Riemannian metric decomposes into three orthogonal components: thescalar curvature, the traceless Ricci curvature tensor, and the Weyl curvature tensor. I will explain in detail therepresentation theory and linear algebra underlying this decomposition. Moreover, we will see that in the specialcases of dimensions 2, 3, 4 one can say more. As an application, I will discuss the Singer-Thorpe Theoremcharacterizing Einstein metrics in 4 dimensions in terms of this decomposition. If time permits (and it may wellpermit, as this will be a 2.5 hour talk with a break midway), I will briefly discuss a generalization of these ideasto G2-geometry in 7 dimensions.
MC 5417