| import OAI.Geometry.IsometricImmersion.Metric |
| import OAI.Geometry.IsometricImmersion.Energy.HeightEnergy |
| import Mathlib.Topology.Algebra.Support |
| import Mathlib.Analysis.Matrix.Normed |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open scoped Matrix.Norms.Elementwise |
| open Filter |
|
|
| namespace SmoothLocal.Geometry |
|
|
| theorem coordPartial_eq_of_eventuallyEq |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f h : Coord → V} {p : Coord} (heq : f =ᶠ[𝓝 p] h) (i : Fin 2) : |
| coordPartial i f p = coordPartial i h p := by |
| unfold coordPartial |
| rw [heq.fderiv_eq] |
|
|
| theorem coordPartial_eventuallyEq |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f h : Coord → V} {p : Coord} (heq : f =ᶠ[𝓝 p] h) (i : Fin 2) : |
| coordPartial i f =ᶠ[𝓝 p] coordPartial i h := by |
| filter_upwards [heq.eventuallyEq_nhds] with q hq |
| exact coordPartial_eq_of_eventuallyEq hq i |
|
|
| def iteratedCoordPartial {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] : |
| List (Fin 2) → (Coord → V) → Coord → V |
| | [], f => f |
| | i :: is, f => coordPartial i (iteratedCoordPartial is f) |
|
|
| theorem iteratedCoordPartial_eventuallyEq |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| {f h : Coord → V} {p : Coord} (heq : f =ᶠ[𝓝 p] h) (dirs : List (Fin 2)) : |
| iteratedCoordPartial dirs f =ᶠ[𝓝 p] iteratedCoordPartial dirs h := by |
| induction dirs with |
| | nil => exact heq |
| | cons i is ih => exact coordPartial_eventuallyEq ih i |
|
|
| theorem iteratedCoordPartial_zero |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| (dirs : List (Fin 2)) : iteratedCoordPartial dirs (0 : Coord → V) = 0 := by |
| induction dirs with |
| | nil => rfl |
| | cons i is ih => |
| change coordPartial i (iteratedCoordPartial is (0 : Coord → V)) = 0 |
| rw [ih] |
| funext p |
| change fderiv ℝ (fun _ : Coord => (0 : V)) p (Pi.single i 1) = 0 |
| simp |
|
|
| theorem eventuallyEq_zero_of_not_mem_tsupport |
| {V : Type*} [Zero V] (f : Coord → V) {p : Coord} (hp : p ∉ tsupport f) : |
| f =ᶠ[𝓝 p] 0 := by |
| filter_upwards [(isClosed_tsupport f).isOpen_compl.mem_nhds hp] with q hq |
| change f q = 0 |
| by_contra hne |
| exact hq (subset_tsupport f hne) |
|
|
| theorem iteratedCoordPartial_zero_of_not_mem_tsupport |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| (f : Coord → V) {p : Coord} (hp : p ∉ tsupport f) (dirs : List (Fin 2)) : |
| iteratedCoordPartial dirs f p = 0 := by |
| have heq := iteratedCoordPartial_eventuallyEq |
| (eventuallyEq_zero_of_not_mem_tsupport f hp) dirs |
| rw [heq.self_of_nhds, iteratedCoordPartial_zero] |
| rfl |
|
|
| theorem metric_coeff_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) (i j : Fin 2) : |
| (fun q => g q i j) =ᶠ[𝓝 p] (fun q => h q i j) := |
| heq.mono fun _ hq => congrFun (congrFun hq i) j |
|
|
| theorem christoffel_eq_of_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) (k i j : Fin 2) : |
| christoffel g k i j p = christoffel h k i j p := by |
| have hval := heq.self_of_nhds |
| have hd (a b c : Fin 2) : coordPartial a (fun q => g q b c) p = |
| coordPartial a (fun q => h q b c) p := |
| coordPartial_eq_of_eventuallyEq (metric_coeff_eventuallyEq heq b c) a |
| simp only [christoffel, inverseMetric, hval, hd] |
|
|
| theorem christoffel_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) (k i j : Fin 2) : |
| christoffel g k i j =ᶠ[𝓝 p] christoffel h k i j := |
| heq.eventuallyEq_nhds.mono fun _ hq => christoffel_eq_of_eventuallyEq hq k i j |
|
|
| theorem riemann_eq_of_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) (l k i j : Fin 2) : |
| riemann g l k i j p = riemann h l k i j p := by |
| have hΓ := christoffel_eq_of_eventuallyEq heq |
| have hd (a b c d : Fin 2) : coordPartial a (christoffel g b c d) p = |
| coordPartial a (christoffel h b c d) p := |
| coordPartial_eq_of_eventuallyEq (christoffel_eventuallyEq heq b c d) a |
| simp only [riemann, hΓ, hd] |
|
|
| theorem gaussianCurvature_eq_of_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) : gaussianCurvature g p = gaussianCurvature h p := by |
| simp only [gaussianCurvature, heq.self_of_nhds, riemann_eq_of_eventuallyEq heq] |
|
|
| theorem covHessian_eq_of_metric_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) (z : Coord → ℝ) : covHessian g z p = covHessian h z p := by |
| ext i j |
| simp only [covHessian, christoffel_eq_of_eventuallyEq heq] |
|
|
| theorem heightEnergy_eq_of_metric_eventuallyEq {g h : MetricField} {p : Coord} |
| (heq : g =ᶠ[𝓝 p] h) (z : Coord → ℝ) : heightEnergy g z p = heightEnergy h z p := by |
| simp only [heightEnergy, covectorNormSq, inverseMetric, heq.self_of_nhds] |
|
|
| theorem metric_add_eventuallyEq_of_not_mem_tsupport (g η : MetricField) |
| {p : Coord} (hp : p ∉ tsupport η) : g + η =ᶠ[𝓝 p] g := by |
| filter_upwards [eventuallyEq_zero_of_not_mem_tsupport η hp] with q hq |
| change g q + η q = g q |
| simp only [hq, Pi.zero_apply, add_zero] |
|
|
| theorem metric_add_all_coordinate_jets (g η : MetricField) |
| {p : Coord} (hp : p ∉ tsupport η) (dirs : List (Fin 2)) : |
| iteratedCoordPartial dirs (g + η) p = iteratedCoordPartial dirs g p := |
| (iteratedCoordPartial_eventuallyEq |
| (metric_add_eventuallyEq_of_not_mem_tsupport g η hp) dirs).self_of_nhds |
|
|
| theorem gaussianCurvature_add_eq_off_tsupport (g η : MetricField) |
| {p : Coord} (hp : p ∉ tsupport η) : |
| gaussianCurvature (g + η) p = gaussianCurvature g p := |
| gaussianCurvature_eq_of_eventuallyEq (metric_add_eventuallyEq_of_not_mem_tsupport g η hp) |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
|
|
