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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | import OAI.Geometry.EinsteinFour.Statement
namespace OAI
/-! Transport the exact metric classification across the classical
constant-rescaling identity for Riemannian distance. -/
namespace PositiveEinsteinFour
open EinsteinFour
open scoped Manifold ContDiff ENNReal
noncomputable section
variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M]
[IsManifold I4 ∞ M]
theorem scaledEDistance_mul (g : SmoothMetric M) (a b : ℝ) (ha : 0 ≤ a)
(x y : M) :
scaledEDistance g (a * b) x y =
ENNReal.ofReal (Real.sqrt a) * scaledEDistance g b x y := by
simp only [scaledEDistance, Real.sqrt_mul ha, ENNReal.ofReal_mul (Real.sqrt_nonneg a),
mul_assoc]
/-- A normalization step needs only its ordinary distance-scaling law;
the final model and positive scale are then transported by this theorem. -/
theorem classified_of_constant_scaling (g h : SmoothMetric M) (c : ℝ) (hc : 0 < c)
(hd : ∀ x y, scaledEDistance h 1 x y = scaledEDistance g c x y)
(hh : Classified h) : Classified g := by
obtain ⟨a, ha, hmodel⟩ := hh
have hscale (x y : M) : scaledEDistance h a x y = scaledEDistance g (a * c) x y := by
calc
_ = scaledEDistance h (a * 1) x y := by rw [mul_one]
_ = ENNReal.ofReal (Real.sqrt a) * scaledEDistance h 1 x y :=
scaledEDistance_mul h a 1 ha.le x y
_ = ENNReal.ofReal (Real.sqrt a) * scaledEDistance g c x y := by rw [hd]
_ = _ := (scaledEDistance_mul g a c ha.le x y).symm
refine ⟨a * c, mul_pos ha hc, ?_⟩
rcases hmodel with hs | hp | hr
· left
obtain ⟨f, hf⟩ := hs
exact ⟨f, fun x y => (hf x y).trans (hscale x y)⟩
· right; left
obtain ⟨f, hf⟩ := hp
exact ⟨f, fun x y => (hf x y).trans (hscale x y)⟩
· right; right
obtain ⟨f, hf⟩ := hr
exact ⟨f, fun x y => (hf x y).trans (hscale x y)⟩
end
end PositiveEinsteinFour
end OAI
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