| 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 |
|
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| end OAI |
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