lemma A.630 Codazzi equation

open in the book · appendices/A-long-proofs.tex:30490 · p. 3096

Rests on

depends_on definition A.625 Four-dimensional extrinsic curvature ¶
- depends_on definition 13.145 Affine connection ¶
  - depends_on definition 13.90 Mixed tensor ¶
    - depends_on definition 13.89 Contravariant and covariant tensors ¶
      - depends_on definition 13.88 General coordinate transformation ¶
    - depends_on definition 13.88 General coordinate transformation ¶ ↺
  - depends_on definition 13.82 Vector field ¶
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
- depends_on equation A.972 eq:app-adm-gauss-codazzi-projector ¶
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 22.29 Poisson bracket ¶
      - depends_on notation 26.1 Weak and strong equality ¶
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - 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-380 ¶

Supports

depends_on theorem 44.42 The constraint equations ¶
- depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
  - depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶
- depends_on theorem 44.46 Maximal globally hyperbolic development; Choquet-Bruhat and Geroch; imported ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Four-dimensional extrinsic curvature	declared	appendices/A-long-proofs.tex:30506
`depends_on`	→	eq:mfd-ricci-identity	declared	appendices/A-long-proofs.tex:30506
`depends_on`	→	Decomposition of $\nabla n$, and the acceleration	declared	appendices/A-long-proofs.tex:30506
`depends_on`	←	The constraint equations	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1724
`proves`	←	app:A-long-proofs@proof-380	declared	appendices/A-long-proofs.tex:30510