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

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Chapter Six

 

Conformal Curvature Tensors

In this chapter we study conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on tensors. We assume throughout this chapter that n ≥ 3.

Let g be a metric on a manifold M. By Theorem 2.9, there is an ambient metric in normal form relative to g, which by Proposition 2.6 we may take to be straight. Such a metric takes the form (3.14) on a neighborhood of × M × {0} in × M × . Equations (3.17) determine the 1-parameter family of metrics ...

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