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.