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I want an expression for the Riemann tensor of the four metric in terms of extrinsic curvature, normal, lie derivative of the normal, etc. The first Einstein-Codacci eq. gives the Riemann tensor of the three metric in terms of the contracted Riemann four tensor and the extrinsic curvature. So this is not the one I want. Does anyone know a relation that I am looking for? And how to derive it?

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See this reference

The formula is on Page 11 (Codazzi Equation) (The definition of the metrics $\gamma$ is on page 7).

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