3 May 2016; IMS (UiTø), RealFag A228. 13:00-14:00.

Eivind Schneider

“Differential invariants of self-dual conformal structures”

Let M be an oriented four dimensional manifold with a pseudo-Riemannian metric g of signature (4,0) or (2,2). For such manifolds the Hodge star is an ivolutive endomorphism on the space of 2-forms. Denoting the Weyl tensor of g by W, we say that M is self-dual if *W=W. Self-duality is invariant with respect to conformal re-scalings, and so is an invariant property of the conformal structure [g].

We give a description of invariants of such self-dual conformal structures with respect to the group of diffeomorphisms Diff(M). This is done in two different ways. First we consider all conformal metrics satisfying *W=W. Locally these are solutions to a system of five differential equations in nine unknown functions, which is then factored by the pseudogroup Diff_{loc}(M). The other method (applicable only in split-signature) uses a normal form of (anti-) self-dual metrics due to Dunajski, Ferapontov and Kruglikov, in which the self-duality equation is written as a system of three differential equations in three unknown functions and we factor this system by its symmetry pseudogroup.

In both approaches we compute the number and the form of differential invariants that separate generic orbits and thus solve the recognition problem for regular self-dual conformal structures.

Joint work with Boris Kruglikov.