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Inhomogeneous ambient metrics. (English) Zbl 1148.53023
Eastwood, Michael (ed.) et al., Symmetries and overdetermined systems of partial differential equations. Proceedings of the IMA summer program, Minneapolis, MN, USA, July 17–August 4, 2006. New York, NY: Springer (ISBN 978-0-387-73830-7/hbk). The IMA Volumes in Mathematics and its Applications 144, 403-420 (2008).
From the authors’ abstract: In this article, we describe a modification to the form of the ambient metric in even dimensions which enables us to obtain invariantly defined, smooth infinite order ‘ambient metrics’. We introduce what we call inhomogeneous ambient metrics, which are formally Ricci-flat and have an asymptotic expansion involving the logarithm of a defining function for the initial surface which is homogeneous of degree 2. Such metrics are themselves no longer homogeneous, and of course are not smooth. However, we are able to define the smooth part of such a metric in an invariant way, and the smooth part is homogeneous and of course smooth and can be used in applications just as the ambient metric itself is used in odd dimensions. A significant difference, though, is that an inhomogeneous ambient metric is no longer uniquely determined up to diffeomorphism by the conformal class $$[g]$$ on $$M$$; there is a family of diffeomorphism classes of inhomogeneous ambient metrics, and therefore of their smooth parts. Upon choosing a metric $$g$$ in the conformal class, this family can be parametrized by the choice of an arbitrary trace-free symmetric $$2$$-tensor field on $$M$$ which we call the ambiguity tensor relative to the representative $$g$$.
We also indicate how inhomogeneous ambient metrics can be used to complete the description of scalar invariants of conformal structures in even dimensions. One can form scalar invariants as Weyl invariants, defined to be linear combinations of complete contractions of covariant derivatives of the curvature tensor of the smooth part of an inhomogeneous ambient metric, and also its volume form and a modified volume form in the case of odd invariants.
For the entire collection see [Zbl 1126.35005].

##### MSC:
 53C20 Global Riemannian geometry, including pinching
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