On the Weyl tensor of a self-dual complex 4-manifold

TL;DR

This paper explores the geometry of self-dual complex 4-manifolds, revealing how the Weyl tensor relates to projective curvature and null-geodesics, and establishing conditions for conformal flatness.

Contribution

It provides a new interpretation of the Weyl tensor as projective curvature on ambitwistor space and links null-geodesics to conformal flatness in self-dual manifolds.

Findings

Weyl tensor interpreted as projective curvature of cone fields

Vanishing Weyl tensor implies existence of certain null-geodesics

Self-dual 4-manifolds with rational null-geodesics are conformally flat

Abstract

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also relate the Cotton-York tensor of an umbilic hypersurface to the Weyl tensor of the ambient. As a consequence, a conformal 3-manifold or a self-dual 4-manifold admitting a rational curve as a null-geodesic is conformally flat. We show that the projective structure of the beta-surfaces of a self-dual manifold is flat.