In conformal geometry, the ambient construction refers to a construction of Charles Fefferman and Robin Graham[1] 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.[2]
A related construction is the tractor bundle.