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695bd70 | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Geometry.SmoothYau.Model
namespace OAI
noncomputable section
open Set Filter
open scoped Topology ContDiff
open Set Filter
open scoped Topology ContDiff
open MvPolynomial
open Set Filter
open scoped ContDiff
open Set Filter
open scoped Topology ContDiff
open Set Filter MvPolynomial
open scoped Topology ContDiff
open Set Filter Function MvPolynomial
open scoped Topology ContDiff
open Set Filter Function MvPolynomial
open scoped Topology ContDiff
open Set Filter
open scoped Topology ContDiff
open Set Filter
open scoped Topology ContDiff
open Set Filter Function
open scoped Topology ContDiff
open Set Filter Function
open scoped Topology ContDiff
open scoped Topology
open Set Filter Manifold Bundle MeasureTheory
open scoped Topology ContDiff ENNReal
namespace YauCounterexamples
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] {M : Type*} [TopologicalSpace M] [ChartedSpace E M]
[IsManifold 𝓘(ℝ, E) ∞ M]
def chartTransition (p q : M) : E → E := (chartAt E p) ∘ (chartAt E q).symm
def coordinateTransitionMatrix (p q : M) (y : E) :
Matrix (CoordIndex E) (CoordIndex E) ℝ :=
LinearMap.toMatrix (Module.finBasis ℝ E) (Module.finBasis ℝ E)
(fderiv ℝ (chartTransition p q) y).toLinearMap
def inverseChartDerivative (p : M) (y : E) : E →L[ℝ] E :=
mfderiv 𝓘(ℝ, E) 𝓘(ℝ, E) (chartAt E p).symm y
omit [FiniteDimensional ℝ E] in
lemma invChart_derivative_change (p q : M) {y : E}
(hy : y ∈ (chartAt E q).target) (hp : (chartAt E q).symm y ∈ (chartAt E p).source) :
(inverseChartDerivative p (chartTransition p q y)).comp
(fderiv ℝ (chartTransition p q) y) =
inverseChartDerivative q y := by
have hpc := mdifferentiable_chart (I := 𝓘(ℝ, E)) p
have hqc := mdifferentiable_chart (I := 𝓘(ℝ, E)) q
have hc := mfderiv_comp y (hpc.mdifferentiableAt hp) (hqc.mdifferentiableAt_symm hy)
rw [mfderiv_eq_fderiv] at hc
change fderiv ℝ (chartTransition p q) y = _ at hc
unfold inverseChartDerivative
erw [hc, ← ContinuousLinearMap.comp_assoc, hpc.symm_comp_deriv hp,
ContinuousLinearMap.id_comp]
lemma coordinateVector_change (p q : M) {y : E}
(hy : y ∈ (chartAt E q).target) (hp : (chartAt E q).symm y ∈ (chartAt E p).source)
(i : CoordIndex E) :
coordinateVector q y i = ∑ k, coordinateTransitionMatrix p q y k i •
coordinateVector p (chartTransition p q y) k := by
have hh := congrArg (fun L : E →L[ℝ] E => L (Module.finBasis ℝ E i))
(invChart_derivative_change p q hy hp)
change _ = coordinateVector q y i at hh
rw [← hh]
change (inverseChartDerivative p (chartTransition p q y))
((fderiv ℝ (chartTransition p q) y) (Module.finBasis ℝ E i)) = _
rw [← (Module.finBasis ℝ E).sum_repr ((fderiv ℝ (chartTransition p q) y) (Module.finBasis ℝ E i))]
simp only [map_sum, map_smul, coordinateTransitionMatrix, LinearMap.toMatrix_apply]
rfl
omit [FiniteDimensional ℝ E] in
private lemma bilinear_sum_matrix {ι : Type*} [Fintype ι]
(B : E →L[ℝ] E →L[ℝ] ℝ) (a : ι → ι → ℝ) (v : ι → E) (i j : ι) :
B (∑ k, a k i • v k) (∑ k, a k j • v k) =
∑ k, ∑ l, a l i * B (v l) (v k) * a k j := by
simp only [map_sum, map_smul, sum_apply, smul_apply, smul_eq_mul, Finset.mul_sum]
apply Finset.sum_congr rfl
intro k _
apply Finset.sum_congr rfl
intro l _
ring
lemma metricCoefficients_change (g : SmoothMetric E M) (p q : M) {y : E}
(hy : y ∈ (chartAt E q).target) (hp : (chartAt E q).symm y ∈ (chartAt E p).source) :
metricCoefficients g q y = (coordinateTransitionMatrix p q y).transpose *
metricCoefficients g p (chartTransition p q y) * coordinateTransitionMatrix p q y := by
ext i j
have he : (chartAt E p).symm (chartTransition p q y) = (chartAt E q).symm y :=
(chartAt E p).left_inv hp
simp only [Matrix.mul_apply, Matrix.transpose_apply, metricCoefficients, Finset.sum_mul]
rw [he]
let B : E →L[ℝ] E →L[ℝ] ℝ := g.inner ((chartAt E q).symm y)
change B (coordinateVector q y i) (coordinateVector q y j) = _
rw [coordinateVector_change p q hy hp i, coordinateVector_change p q hy hp j]
change B (∑ k : CoordIndex E, coordinateTransitionMatrix p q y k i •
(coordinateVector p (chartTransition p q y) k : E))
(∑ k : CoordIndex E, coordinateTransitionMatrix p q y k j •
(coordinateVector p (chartTransition p q y) k : E)) = _
exact bilinear_sum_matrix B (coordinateTransitionMatrix p q y)
(fun k => (coordinateVector p (chartTransition p q y) k : E)) i j
end YauCounterexamples
end
end OAI
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