File size: 2,569 Bytes
59d1c80 | 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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Geometry.IsometricImmersion.CoordinateMain
import OAI.Geometry.IsometricImmersion.LocalDomains
namespace OAI
noncomputable section
open Set TopologicalSpace
open scoped ContDiff Topology Manifold Matrix
namespace SmoothLocal.Geometry
def coordinateSquareOpen : Opens Coord :=
⟨square, SmoothLocal.ODE.coordinate_square_isOpen⟩
theorem zero_mem_coordinateSquareOpen : (0 : Coord) ∈ coordinateSquareOpen := by
intro i
norm_num
theorem relativelyOpen_square_iff {U : Set Coord} (hUS : U ⊆ square) :
IsOpen ((Subtype.val : coordinateSquareOpen → Coord) ⁻¹' U) ↔ IsOpen U := by
constructor
· intro h
have he : (Subtype.val : coordinateSquareOpen → Coord) ''
(Subtype.val ⁻¹' U) = U := by
ext p
constructor
· rintro ⟨q, hq, rfl⟩
exact hq
· intro hp
exact ⟨⟨p, hUS hp⟩, hp, rfl⟩
have ho := coordinateSquareOpen.isOpen.isOpenMap_subtype_val _ h
exact (congrArg IsOpen he).mp ho
· intro h
exact h.preimage continuous_subtype_val
theorem counterexampleMetric_local_smoothPositive :
LocalSmoothPositive coordinateSquareOpen (fun p => counterexampleMetric p) :=
(localSmoothPositive_restrict_iff coordinateSquareOpen counterexampleMetric).mpr
counterexampleMetric_smoothPositive
theorem counterexampleMetric_no_subtype_isometric_immersion
(U : Opens Coord) (hUS : (U : Set Coord) ⊆ square)
(h0 : (0 : Coord) ∈ U) (F : U → Ambient) :
¬ LocalIsometric counterexampleMetric U F := by
intro hF
exact counterexampleMetric_no_local_isometric_immersion U.isOpen hUS h0
(openDomainExtend U F 0) ((localIsometric_extend_iff _ U F 0).mp hF)
theorem exists_local_metric_without_local_immersion :
∃ g : coordinateSquareOpen → Matrix (Fin 2) (Fin 2) ℝ,
LocalSmoothPositive coordinateSquareOpen g ∧
∀ U : Opens Coord, ∀ hUS : (U : Set Coord) ⊆ square,
(0 : Coord) ∈ U → ∀ F : U → Ambient,
¬ (ContMDiff 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) ∞ F ∧
∀ p : U, ∀ v w : Coord,
(inner ℝ : Ambient → Ambient → ℝ) (mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p v)
(mfderiv 𝓘(ℝ, Coord) 𝓘(ℝ, Ambient) F p w) =
v ⬝ᵥ (g ⟨p, hUS p.property⟩ *ᵥ w)) := by
refine ⟨fun p => counterexampleMetric p, counterexampleMetric_local_smoothPositive, ?_⟩
intro U hUS h0 F hF
exact counterexampleMetric_no_subtype_isometric_immersion U hUS h0 F hF
end SmoothLocal.Geometry
end
end OAI
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