lemma A.631 The normal–normal Ricci contraction

open in the book · appendices/A-long-proofs.tex:30572 · p. 3097

Rests on

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 13.44 Atlas ¶
  - depends_on definition 13.84 Tensor ¶
    - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶
    - 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 13.89 Contravariant and covariant tensors ¶
      - 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 13.147 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 definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-64 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
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 definition 13.145 Affine connection ¶ ↺
  - depends_on equation A.972 eq:app-adm-gauss-codazzi-projector ¶
- 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 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 ¶

Supports

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

Neighborhood

Every logical edge within two steps of this node.

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

Edges

type	direction	node	provenance	where
`depends_on`	→	Contractions	declared	appendices/A-long-proofs.tex:30581
`depends_on`	→	eq:mfd-ricci-identity	declared	appendices/A-long-proofs.tex:30581
`depends_on`	→	Decomposition of $\nabla n$, and the acceleration	declared	appendices/A-long-proofs.tex:30581
`depends_on`	←	The $3+1$ identity for the curvature scalar	declared	appendices/A-long-proofs.tex:30650
`proves`	←	app:A-long-proofs@proof-381	declared	appendices/A-long-proofs.tex:30585