import OAI.Geometry.IsometricImmersion.Metric
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Tactic.FinCases

namespace OAI

noncomputable section
open scoped ContDiff BigOperators Matrix

namespace SmoothLocal.Geometry

variable {g : MetricField} {U : Set Coord}

theorem metricDet_ne_zero (hg : SmoothPositiveOn g U) {p : Coord} (hp : p ∈ U) :
    (g p).det ≠ 0 :=
  ((Matrix.isUnit_iff_isUnit_det _).mp (hg.2 p hp).isUnit).ne_zero

theorem metricDet_contDiffOn (hg : SmoothPositiveOn g U) :
    ContDiffOn ℝ ∞ (fun p => (g p).det) U := by
  simpa only [Matrix.det_fin_two] using
    ((hg.1 0 0).mul (hg.1 1 1)).sub ((hg.1 0 1).mul (hg.1 1 0))

theorem metricAdjugate_contDiffOn (hg : SmoothPositiveOn g U) (i j : Fin 2) :
    ContDiffOn ℝ ∞ (fun p => (g p).adjugate i j) U := by
  fin_cases i <;> fin_cases j
  · simpa [Matrix.adjugate_fin_two] using hg.1 1 1
  · simpa [Matrix.adjugate_fin_two] using (hg.1 0 1).neg
  · simpa [Matrix.adjugate_fin_two] using (hg.1 1 0).neg
  · simpa [Matrix.adjugate_fin_two] using hg.1 0 0

theorem inverseMetric_contDiffOn (hg : SmoothPositiveOn g U) (i j : Fin 2) :
    ContDiffOn ℝ ∞ (fun p => inverseMetric g p i j) U := by
  have hi : ContDiffOn ℝ ∞ (fun p => ((g p).det)⁻¹) U :=
    (metricDet_contDiffOn hg).inv (fun p hp => metricDet_ne_zero hg hp)
  have ha := metricAdjugate_contDiffOn hg i j
  simpa [inverseMetric, Matrix.inv_def, Ring.inverse_eq_inv, smul_eq_mul] using hi.mul ha

theorem christoffel_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U)
    (k i j : Fin 2) :
    ContDiffOn ℝ ∞ (christoffel g k i j) U := by
  have hs : ContDiffOn ℝ ∞ (fun p => ∑ 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)) U := by
    apply ContDiffOn.sum
    intro l hl
    exact (inverseMetric_contDiffOn hg k l).mul
      (((partial_contDiffOn (hg.1 j l) hU i).add
        (partial_contDiffOn (hg.1 i l) hU j)).sub
        (partial_contDiffOn (hg.1 i j) hU l))
  exact contDiffOn_const.mul hs

theorem covHessian_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U)
    {z : Coord → ℝ} (hz : ContDiffOn ℝ ∞ z U) (i j : Fin 2) :
    ContDiffOn ℝ ∞ (fun p => covHessian g z p i j) U := by
  have hs : ContDiffOn ℝ ∞ (fun p => ∑ k,
      christoffel g k i j p * coordPartial k z p) U := by
    apply ContDiffOn.sum
    intro k hk
    exact (christoffel_contDiffOn hg hU k i j).mul (partial_contDiffOn hz hU k)
  exact (partial_contDiffOn (partial_contDiffOn hz hU j) hU i).sub hs

theorem covectorNormSq_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U)
    {z : Coord → ℝ} (hz : ContDiffOn ℝ ∞ z U) :
    ContDiffOn ℝ ∞ (covectorNormSq g z) U := by
  apply ContDiffOn.sum
  intro i hi
  apply ContDiffOn.sum
  intro j hj
  exact ((inverseMetric_contDiffOn hg i j).mul (partial_contDiffOn hz hU i)).mul
    (partial_contDiffOn hz hU j)

theorem riemann_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U)
    (l k i j : Fin 2) :
    ContDiffOn ℝ ∞ (riemann g l k i j) U := by
  have hs : ContDiffOn ℝ ∞ (fun 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)) U := by
    apply ContDiffOn.sum
    intro m hm
    exact ((christoffel_contDiffOn hg hU m j k).mul
      (christoffel_contDiffOn hg hU l i m)).sub
        ((christoffel_contDiffOn hg hU m i k).mul
          (christoffel_contDiffOn hg hU l j m))
  exact ((partial_contDiffOn (christoffel_contDiffOn hg hU l j k) hU i).sub
    (partial_contDiffOn (christoffel_contDiffOn hg hU l i k) hU j)).add hs

theorem gaussianCurvature_contDiffOn (hg : SmoothPositiveOn g U) (hU : IsOpen U) :
    ContDiffOn ℝ ∞ (gaussianCurvature g) U := by
  have hn : ContDiffOn ℝ ∞ (fun p => ∑ l, g p 0 l * riemann g l 1 0 1 p) U := by
    apply ContDiffOn.sum
    intro l hl
    exact (hg.1 0 l).mul (riemann_contDiffOn hg hU l 1 0 1)
  exact hn.div (metricDet_contDiffOn hg) (fun p hp => metricDet_ne_zero hg hp)

end SmoothLocal.Geometry

end

end OAI