example 13.144 The two Killing tensors every metric carries

Rests on

• depends_on definition 13.142 Killing tensor ¶
  • depends_on definition 13.139 Killing vector ¶
    • depends_on definition 13.120 Isometry ¶
      • 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 equation 13.267 eq:mfd-lie-def ¶
  • 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 13.146 Parallel transport and autoparallels ¶
      • depends_on definition 13.145 Affine connection ¶ ↺
  • depends_on definition 13.91 Symmetric and antisymmetric parts ¶
• depends_on definition 13.149 Metric compatibility ¶ ↺
• depends_on proposition 13.140 Killing's equation ¶
  • depends_on definition 13.139 Killing vector ¶ ↺
  • depends_on definition 13.149 Metric compatibility ¶ ↺
  • depends_on proposition 13.127 Component formulas ¶
    • depends_on definition 13.89 Contravariant and covariant tensors ¶ ↺
    • depends_on definition 13.90 Mixed tensor ¶ ↺
    • depends_on equation 13.267 eq:mfd-lie-def ¶ ↺
    • proves proof ch:11-manifolds-tensors-curvature@proof-28 ¶
  • 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 ¶
      • depends_on definition 13.145 Affine connection ¶ ↺
    • proves proof ch:11-manifolds-tensors-curvature@proof-41 ¶
  • 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 ¶
      • 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 ¶
  • proves proof ch:11-manifolds-tensors-curvature@proof-37 ¶

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