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The Ambient Metric (AM-178) by C. Robin Graham, Charles Fefferman

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Chapter Seven

 

Conformally Flat and Conformally Einstein Spaces

If n is odd, an ambient metric in normal form is uniquely determined to infinite order by (M, g). Theorem 3.9 shows that within the family of all formal solutions, this one is distinguished by the vanishing of the figure coefficient, or equivalently by the condition that it be smooth. In the Poincaré realization, the corresponding condition is that the expansion have no rn term, or equivalently that it be even. The fact that there is such a normalization giving rise to a unique diffeomorphism class of ambient metrics associated to the conformal manifold (M, [ g]) is crucial for the applications ...

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