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Chapter 6. Conformal Curvature Tensors - Pg. 56

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 cur- vature tensor of an ambient metric in normal form relative to g. Their trans- formation 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 R + × M × {0} in R + × M × R. Equations (3.17) determine the 1-parameter family of metrics g ij (x, ) on M in terms of the initial metric to infinite order n/2 for n odd and modulo O( n/2 ) for n even, except that also g ij g ij | =0 is determined for n even. Each of the determined Taylor coefficients is a natural invariant of the initial metric g.