proposition 13.148 prop:mfd-torsion-tensor

Rests on

• 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 13.88 General coordinate transformation ¶ ↺
• depends_on definition 13.147 Torsion ¶
  • depends_on definition 13.145 Affine connection ¶
    • depends_on definition 13.90 Mixed tensor ¶ ↺
    • depends_on definition 13.82 Vector field ¶
      • depends_on definition 13.81 Vector on a manifold ¶
        • depends_on definition 13.71 Differentiable curve ¶
        • depends_on definition 13.45 Differentiable map on a topological space ¶
        • depends_on definition 13.48 Differentiable manifold ¶
• depends_on equation 13.296 eq:mfd-gamma-transformation ¶
• 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 ¶
• proves proof ch:11-manifolds-tensors-curvature@proof-40 ¶

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