definition A.625 Four-dimensional extrinsic curvature

open in the book · appendices/A-long-proofs.tex:30212 · p. 3093

ground object -- no derivation owed

Rests on

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 13.48 Differentiable manifold ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
depends_on equation A.972 eq:app-adm-gauss-codazzi-projector ¶

Supports

depends_on lemma A.630 Codazzi equation ¶
- 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 ¶ ↺
depends_on lemma A.628 Gauss equation, arbitrary signature ¶
- depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶
- depends_on theorem 44.42 The constraint equations ¶ ↺
depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
- depends_on lemma A.630 Codazzi equation ¶ ↺
- depends_on lemma A.631 The normal–normal Ricci contraction ¶
  - depends_on theorem A.632 The $3+1$ identity for the curvature scalar ¶ ↺

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`	→	Affine connection	declared	appendices/A-long-proofs.tex:30222
`depends_on`	→	eq:app-adm-gauss-codazzi-projector	declared	appendices/A-long-proofs.tex:30222
`depends_on`	←	Codazzi equation	declared	appendices/A-long-proofs.tex:30506
`depends_on`	←	Gauss equation, arbitrary signature	declared	appendices/A-long-proofs.tex:30400
`depends_on`	←	Decomposition of $\nabla n$, and the acceleration	declared	appendices/A-long-proofs.tex:30241