definition 13.111 Linearised curvature of a symmetric field

open in the book · parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5274 · p. 517

ground object -- no derivation owed

Rests on

depends_on definition 13.91 Symmetric and antisymmetric parts ¶
depends_on proposition 13.154 Symmetries of the curvature ¶
- depends_on definition 13.149 Metric compatibility ¶
  - 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 13.117 Metric tensor of signature $(p,q)$ ¶
    - depends_on definition 13.48 Differentiable manifold ¶
      - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.84 Tensor ¶
      - depends_on definition 13.83 Covector ¶
      - depends_on definition 13.81 Vector on a manifold ¶ ↺
  - depends_on definition 13.146 Parallel transport and autoparallels ¶
    - depends_on definition 13.145 Affine connection ¶ ↺
- depends_on proposition 13.157 Bianchi identities ¶
  - depends_on equation 13.246 eq:mfd-leibniz-forms ¶
  - depends_on equation 13.245 eq:mfd-nilpotency ¶
  - depends_on theorem 13.156 Cartan structure equations ¶
    - depends_on definition 13.103 Exterior derivative ¶
      - depends_on definition 13.98 $k$-form ¶
      - depends_on definition 13.102 Wedge product ¶
    - depends_on definition 13.147 Torsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
    - depends_on definition 13.122 Vielbein ¶
      - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
      - depends_on equation 13.1 eq:mfd-eta ¶
    - depends_on definition 13.102 Wedge product ¶ ↺
    - depends_on equation 13.303 eq:mfd-vielbein-postulate ¶
    - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - depends_on definition 13.145 Affine connection ¶ ↺
      - depends_on definition 13.147 Torsion ¶ ↺
      - depends_on proposition 7.105 Clairaut–Schwarz ¶
      - proves proof ch:11-manifolds-tensors-curvature@proof-42 ¶
    - proves proof ch:11-manifolds-tensors-curvature@proof-45 ¶
  - proves proof ch:11-manifolds-tensors-curvature@proof-46 ¶
- depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
  - depends_on definition 13.145 Affine connection ¶ ↺
  - depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶ ↺
  - depends_on definition 13.149 Metric compatibility ¶ ↺
  - depends_on definition 13.147 Torsion ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
- depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
- proves proof ch:11-manifolds-tensors-curvature@proof-43 ¶

Supports

depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶

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`	→	Symmetric and antisymmetric parts	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5290
`depends_on`	→	Symmetries of the curvature	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5290
`depends_on`	←	Symmetric analogue of the converse Poincaré lemma	declared	parts/02-mathematical-methods/11-manifolds-tensors-curvature.tex:5306