# Khavkine I. Compatibility complexes for the Killing equation (abstract)

Speaker: Igor Khavkine

Title: Compatibility complexes for the Killing equation

Abstract:
The Killing operator on a (pseudo-)Riemannian geometry $(M,g)$ is $K_{ab}[v] = \nabla_a v_b + \nabla_b v_a$. The Killing equation $K[v] = 0$ is an overdetermined PDE and we will consider its compatibility complex $K_i$ ($i\ge 0$), where $K_0 = K$ and any differential operator $C$ satisfying $C\circ K_i=0$ must factor as $C = C' \circ K_{i+1}$, for some differential operator $C'$. Relying on the "finite-type" property of K, I will discuss a practical construction of such a compatibility complex on geometries of sub-maximal symmetry, with examples coming from General Relativity. Prior to this work, there were very few examples with the full compatibility complex $K_i$ or even just $K_1$ known.

Slides: KhavkineTrieste2018slides.pdf

References:
arXiv:1805.03751

Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.