theorem 13.112 Symmetric analogue of the converse Poincaré lemma

Rests on

• depends_on definition 13.111 Linearised curvature of a symmetric field ¶
  • 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 13.82 Vector field ¶
      • depends_on definition 13.117 Metric tensor of signature $(p,q)$ ¶
        • depends_on definition 13.48 Differentiable manifold ¶
        • depends_on definition 13.84 Tensor ¶
      • 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 13.147 Torsion ¶
        • depends_on definition 13.122 Vielbein ¶
        • 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 ¶
        • 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 ¶
• depends_on proposition 7.105 Clairaut–Schwarz ¶
  • depends_on definition 7.20 Continuity at a point ¶
    • depends_on definition 7.16 Limit ¶
      • depends_on definition 7.2 Absolute value ¶
      • depends_on definition 7.9 Real function ¶
        • depends_on definition 3.43 Map ¶
    • 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 definition 7.11 Sum of functions ¶
        • depends_on definition 7.9 Real function ¶ ↺
      • depends_on equation 7.14 eq:ana-derivh ¶
      • depends_on proposition 7.6 Algebra of limits ¶
        • depends_on definition 7.4 Convergence ¶
        • depends_on proposition 7.3 Triangle inequality ¶
        • proves proof ch:05-real-analysis@proof-3 ¶
      • proves proof ch:05-real-analysis@proof-13 ¶
    • depends_on theorem 7.34 Rolle ¶
      • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
        • depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
        • proves proof ch:05-real-analysis@proof-17 ¶
      • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on proposition 7.22 Sequential characterization ¶
        • depends_on theorem 7.7 Bolzano–Weierstrass ¶
        • proves proof ch:05-real-analysis@proof-9 ¶
      • proves proof ch:05-real-analysis@proof-18 ¶
    • proves proof ch:05-real-analysis@proof-19 ¶
  • proves proof ch:05-real-analysis@proof-64 ¶
• depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
  • depends_on definition 13.105 Closed form ¶
    • depends_on definition 13.103 Exterior derivative ¶ ↺
    • depends_on definition 13.98 $k$-form ¶
      • depends_on definition 13.83 Covector ¶
        • depends_on definition 13.81 Vector on a manifold ¶
      • depends_on definition 13.84 Tensor ¶ ↺
  • depends_on definition 13.106 Exact form ¶
    • depends_on definition 13.103 Exterior derivative ¶ ↺
    • depends_on definition 13.98 $k$-form ¶ ↺
  • depends_on definition 13.108 Star-shaped domain ¶
    • depends_on definition 6.2 Open set ¶
      • depends_on definition 6.1 Topological space ¶
        • depends_on definition 3.31 Empty set ¶
        • depends_on definition 3.35 Union, intersection, difference ¶
        • depends_on definition 3.30 Subset ¶
        • depends_on equation 3.51 eq:set-indexed ¶
    • depends_on equation 6.12 eq:top-euclidean-metric ¶
  • proves proof ch:11-manifolds-tensors-curvature@proof-22 ¶
• proves proof ch:11-manifolds-tensors-curvature@proof-23 ¶

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