lemma A.628 Gauss equation, arbitrary signature

open in the book · appendices/A-long-proofs.tex:30388 · p. 3095

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 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 ¶

Supports

depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
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:30400
`depends_on`	→	eq:app-adm-gauss-codazzi-D	declared	appendices/A-long-proofs.tex:30400
`depends_on`	→	eq:mfd-ricci-identity	declared	appendices/A-long-proofs.tex:30400
`depends_on`	←	The $3+1$ identity for the curvature scalar	declared	appendices/A-long-proofs.tex:30650
`depends_on`	←	The constraint equations	declared	parts/05-general-relativity-cosmology/03-einstein-field-equations.tex:1724
`proves`	←	app:A-long-proofs@proof-379	declared	appendices/A-long-proofs.tex:30404