theorem A.632 The $3+1$ identity for the curvature scalar

Rests on

• depends_on lemma A.627 The dictionary between $\mathcal{K},\gamma$ and $K,h$ ¶
  • depends_on definition 13.153 Contractions ¶
    • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
      • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.47 Differentiable structure ¶
      • depends_on definition 13.84 Tensor ¶
        • depends_on definition 13.83 Covector ¶
        • depends_on definition 13.81 Vector on a manifold ¶
    • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      • depends_on definition 13.145 Affine connection ¶
        • depends_on definition 13.90 Mixed tensor ¶
        • depends_on definition 13.82 Vector field ¶
      • depends_on definition 13.147 Torsion ¶
        • depends_on definition 13.145 Affine connection ¶ ↺
      • depends_on proposition 7.105 Clairaut–Schwarz ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-64 ¶
      • proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
  • depends_on equation A.974 eq:app-adm-gauss-codazzi-gamma-h ¶
  • depends_on equation 26.79 eq:cd-adm-extrinsic ¶
  • proves proof app:A-long-proofs@proof-378 ¶
• depends_on lemma A.628 Gauss equation, arbitrary signature ¶
  • depends_on definition A.625 Four-dimensional extrinsic curvature ¶
    • depends_on definition 13.145 Affine connection ¶ ↺
    • depends_on equation A.972 eq:app-adm-gauss-codazzi-projector ¶
  • depends_on equation A.973 eq:app-adm-gauss-codazzi-D ¶
  • depends_on equation 13.305 eq:mfd-ricci-identity ¶
  • proves proof app:A-long-proofs@proof-379 ¶
• depends_on lemma A.631 The normal–normal Ricci contraction ¶
  • depends_on definition 13.153 Contractions ¶ ↺
  • depends_on equation 13.305 eq:mfd-ricci-identity ¶ ↺
  • depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
    • depends_on definition A.625 Four-dimensional extrinsic curvature ¶ ↺
    • depends_on equation 13.295 eq:mfd-covariant-derivative ¶
    • depends_on lemma A.623 The adapted frame of the $3+1$ split ¶
      • depends_on definition 26.40 The $3+1$ decomposition ¶
        • depends_on definition 43.2 Einstein–Hilbert action ¶
        • depends_on notation 26.28 Field systems ¶
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      • depends_on equation 26.78 eq:cd-adm-metric ¶
      • proves proof app:A-long-proofs@proof-376 ¶
    • proves proof app:A-long-proofs@proof-377 ¶
  • proves proof app:A-long-proofs@proof-381 ¶
• proves proof app:A-long-proofs@proof-382 ¶

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