Differential Geometry Working Seminar

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

"Relation between self-duality and conformal structures in dimension 4"

We present the full details of the following classical fact, which is well-known ‘folklore’ but not easy to find in the literature.  Let $V$ be an oriented 4-dimensional real vector space. There is a symmetric bilinear form $Q$ on the 6-dimensional vector space $W = \Lambda^2 V^*$, which is well-defined up to scale. This bilinear form has signature $(3,3)$. We show that there is a one-to-one correspondence between conformal classes of metrics on V (that is, inner products up to positive scale) and maximal spacelike (positive-definite) subspaces of $W$. If time permits, we relate these ideas to the hypersymplectic structures that Amanda talked about last week.

MC 5479