definition 13.88 General coordinate transformation

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• depends_on definition 13.89 Contravariant and covariant tensors ¶
  • depends_on definition 13.90 Mixed tensor ¶
    • depends_on definition 13.145 Affine connection ¶
      • depends_on definition A.625 Four-dimensional extrinsic curvature ¶
        • depends_on lemma A.630 Codazzi equation ¶
        • depends_on lemma A.628 Gauss equation, arbitrary signature ¶
        • depends_on lemma A.626 Decomposition of $\nabla n$, and the acceleration ¶
      • depends_on definition A.624 Projector and induced metric ¶
      • depends_on definition 13.77 Curvature vector ¶
        • depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶
      • depends_on definition 13.149 Metric compatibility ¶
        • depends_on definition 13.142 Killing tensor ¶
        • depends_on example 13.144 The two Killing tensors every metric carries ¶
        • depends_on lemma 44.8 Palatini identity ¶
        • depends_on proposition 44.30 The cosmological term ¶
        • depends_on proposition 13.78 The curvature vector is orthogonal to the tangent ¶ ↺
        • depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶
        • depends_on proposition 13.140 Killing's equation ¶
        • depends_on proposition 13.154 Symmetries of the curvature ¶
        • depends_on theorem 44.23 Contracted Bianchi identity ¶
        • depends_on theorem 13.79 Frenet–Serret equations, covariant form ¶
        • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶
      • depends_on definition 13.146 Parallel transport and autoparallels ¶
        • depends_on definition 13.149 Metric compatibility ¶ ↺
        • depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
        • depends_on proposition 13.141 The invariant of a Killing vector along a geodesic ¶ ↺
        • depends_on remark 29.62 The same rate read as a holonomy ¶
        • depends_on theorem 13.143 Killing tensors and geodesic invariants ¶
      • depends_on definition 13.147 Torsion ¶
        • depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
        • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
        • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
        • depends_on theorem 13.156 Cartan structure equations ¶
      • depends_on theorem 13.150 Levi-Civita connection and contorsion ¶ ↺
      • depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶ ↺
    • depends_on proposition 13.127 Component formulas ¶
      • depends_on lemma A.75 Differentiating a pullback along a flow ¶
        • depends_on lemma A.77 Poincaré lemma, converse form ¶
      • depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶
      • depends_on proposition 13.128 Cartan's magic formula ¶
        • depends_on lemma A.77 Poincaré lemma, converse form ¶ ↺
        • depends_on proposition 13.129 Commutation of Lie derivative and interior product ¶ ↺
        • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      • depends_on proposition 13.140 Killing's equation ¶ ↺
      • depends_on theorem 44.19 Covariant conservation of stress–energy ¶
        • depends_on proposition 44.20 Perfect-fluid equations of motion ¶
    • depends_on proposition 13.148 prop:mfd-torsion-tensor ¶ ↺
  • depends_on definition 13.92 Tensor density ¶
    • depends_on example 13.94 The two densities this chapter uses ¶
      • depends_on lemma 14.62 The Levi-Civita symbol is invariant ¶
        • depends_on definition 29.4 Angular velocity ¶
        • depends_on lemma 14.64 Properties of $W$ ¶
    • depends_on proposition 13.93 Algebra of densities ¶
      • depends_on example 13.94 The two densities this chapter uses ¶ ↺
  • depends_on proposition 13.127 Component formulas ¶ ↺
• depends_on definition 13.90 Mixed tensor ¶ ↺
• depends_on definition 13.92 Tensor density ¶ ↺
• depends_on proposition 13.75 Invariance of the length ¶