Ambient construction
Open-knowledge reference entry
In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham for which a conformal manifold of dimension n is realized (ambiently) as the boundary of a certain Poincaré manifold, or alternatively as the celestial sphere of a certain pseudo-Riemannian manifold.
The ambient construction is canonical in the sense that it is performed only using the conformal class of the metric: it is conformally invariant. However, the construction only works asymptotically, up to a certain order of approximation. There is, in general, an obstruction to continuing this extension past the critical order. The obstruction itself is of tensorial character, and is known as the (conformal) obstruction tensor. It is, along with the Weyl tensor, one of the two primitive invariants in conformal differential geometry.
Aside from the obstruction tensor, the ambient construction can be used to define a class of conformally invariant differential operators known as the GJMS operators.
A related construction is the tractor bundle.
Overview
The model flat geometry for the ambient construction is the future null cone in Minkowski space, with the origin deleted. The celestial sphere at infinity is the conformal manifold M, and the null rays in the cone determine a line bundle over M. Moreover, the null cone carries a metric which degenerates in the direction of the generators of the cone.
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