File size: 3,025 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 69 70 71 72 73 74 75 76 | 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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