lemma A.640 Variation of the integrated three-curvature

open in the book · appendices/A-long-proofs.tex:31168 · p. 3103

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 13.299 eq:mfd-gamma-decomposition ¶
depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
proves proof app:A-long-proofs@proof-386 ¶

Supports

depends_on theorem A.641 The normal–normal bracket ¶
- depends_on corollary A.642 The constraints are first class ¶

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`	→	Contractions	declared	appendices/A-long-proofs.tex:31180
`depends_on`	→	eq:mfd-gamma-decomposition	declared	appendices/A-long-proofs.tex:31180
`depends_on`	→	Riemann tensor; Ricci identity with torsion	declared	appendices/A-long-proofs.tex:31180
`depends_on`	←	The normal–normal bracket	declared	appendices/A-long-proofs.tex:31247
`proves`	←	app:A-long-proofs@proof-386	declared	appendices/A-long-proofs.tex:31184