 Gauss–Codazzi equations for a spatial hypersurface

ID: gauss-codazzi-equations-for-a-spatial-hypersurface

With signature

(-, +, +, +)

n

and

K_{ab} = - P^{c}_{a} P^{d}_{b} \nabla_{c} n_{d}

(⊥ R)_{ab c d} = R_{ab c d} + K_{a c} K_{d b} - K_{a d} K_{c b}

and

(⊥ R)_{ab c d} n^{d} = - D_{a} K_{b c} + D_{b} K_{a c}

, where in the second expression only the first three indices are projected. The intrinsic curvature and spatial covariant derivative use the spatial induced metric. The normal is timelike, so the normal-sign convention differs from a spacelike normal in a positive-definite ambient metric.

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