| import OAI.Geometry.IsometricImmersion.Calculus.CoordinateDerivatives |
| import Mathlib.LinearAlgebra.Matrix.PosDef |
| import Mathlib.LinearAlgebra.Matrix.NonsingularInverse |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Matrix ContDiff |
| namespace SmoothLocal.Geometry |
|
|
| abbrev Ambient := EuclideanSpace ℝ (Fin 3) |
| abbrev MetricField := Coord → Matrix (Fin 2) (Fin 2) ℝ |
|
|
| def square : Set Coord := {p | ∀ i, p i ∈ Set.Ioo (-1 : ℝ) 1} |
|
|
| def SmoothPositiveOn (g : MetricField) (U : Set Coord) : Prop := |
| (∀ i j, ContDiffOn ℝ ∞ (fun p => g p i j) U) ∧ ∀ p ∈ U, (g p).PosDef |
|
|
| def IsometricOn (g : MetricField) (F : Coord → Ambient) (U : Set Coord) : Prop := |
| ContDiffOn ℝ ∞ F U ∧ ∀ p ∈ U, ∀ v w : Coord, |
| inner ℝ (fderiv ℝ F p v) (fderiv ℝ F p w) = v ⬝ᵥ (g p *ᵥ w) |
|
|
| def inverseMetric (g : MetricField) (p : Coord) := (g p)⁻¹ |
|
|
| def christoffel (g : MetricField) (k i j : Fin 2) (p : Coord) : ℝ := |
| (1 / 2 : ℝ) * ∑ l, inverseMetric g p k l * |
| (coordPartial i (fun q => g q j l) p + coordPartial j (fun q => g q i l) p - |
| coordPartial l (fun q => g q i j) p) |
|
|
| def covHessian (g : MetricField) (z : Coord → ℝ) (p : Coord) : |
| Matrix (Fin 2) (Fin 2) ℝ := fun i j => |
| coordPartial i (coordPartial j z) p - ∑ k, christoffel g k i j p * coordPartial k z p |
|
|
| def covectorNormSq (g : MetricField) (z : Coord → ℝ) (p : Coord) : ℝ := |
| ∑ i, ∑ j, inverseMetric g p i j * coordPartial i z p * coordPartial j z p |
|
|
| def riemann (g : MetricField) (l k i j : Fin 2) (p : Coord) : ℝ := |
| coordPartial i (christoffel g l j k) p - coordPartial j (christoffel g l i k) p + |
| ∑ m, (christoffel g m j k p * christoffel g l i m p - |
| christoffel g m i k p * christoffel g l j m p) |
|
|
| def gaussianCurvature (g : MetricField) (p : Coord) : ℝ := |
| (∑ l, g p 0 l * riemann g l 1 0 1 p) / (g p).det |
|
|
| def height (F : Coord → Ambient) (e : Ambient) : Coord → ℝ := |
| fun p => inner ℝ (F p) e |
|
|
| def secondFundamental (F : Coord → Ambient) (n : Ambient) (p : Coord) : |
| Matrix (Fin 2) (Fin 2) ℝ := |
| fun i j => inner ℝ (coordPartial i (coordPartial j F) p) n |
|
|
| theorem isometricOn_injective_differential {g : MetricField} {F : Coord → Ambient} |
| {U : Set Coord} (hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) |
| {p : Coord} (hp : p ∈ U) : Function.Injective (fderiv ℝ F p) := by |
| intro v w hvw |
| have hL : fderiv ℝ F p (v - w) = 0 := by simp [map_sub, hvw] |
| have hv : v - w = 0 := by |
| by_contra hne |
| have hpos := (hg.2 p hp).dotProduct_mulVec_pos hne |
| have heq := hF.2 p hp (v - w) (v - w) |
| rw [hL, inner_zero_left] at heq |
| have hpos' : 0 < (v - w) ⬝ᵥ (g p *ᵥ (v - w)) := by simpa using hpos |
| exact (ne_of_gt hpos') heq.symm |
| exact sub_eq_zero.mp hv |
|
|
| def MainTarget : Prop := |
| ∃ g : MetricField, SmoothPositiveOn g square ∧ |
| ∀ U : Set Coord, IsOpen U → U ⊆ square → (0 : Coord) ∈ U → |
| ∀ F : Coord → Ambient, ¬ IsometricOn g F U |
|
|
| end SmoothLocal.Geometry |
|
|
| end |
|
|
| end OAI |
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|
