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def definition def or abbrev

ricci_tensor

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formal statement (Lean)

  51noncomputable def ricci_tensor (g : MetricTensor) : BilinearForm :=

proof body

Definition body.

  52  fun x _ low =>
  53    let mu := low 0
  54    let nu := low 1
  55    Finset.univ.sum (fun (rho : Fin 4) =>
  56      riemann_tensor g x (fun _ => rho) (fun i => if i.val = 0 then mu else if i.val = 1 then rho else nu))
  57
  58/-- **THEOREM (Riemann Antisymmetry)**: The Riemann tensor is antisymmetric in its last two indices.
  59    R^ρ_{σμν} = -R^ρ_{σνμ}
  60
  61    This follows directly from the definition of the Riemann tensor in terms of
  62    Christoffel symbols. -/

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