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import OAI.Analysis.BiholderTransport.Geodesics.MinimizingChartParameters
namespace OAI
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff
namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteNormalInjective (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end WeakMTWTransport
end
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
def minimizingEndpointChartParameters (a c : M) (K L : Set M) :
Set (Model n×(Model n×Model n)) :=
(fun z : TangentBundle 𝓘(ℝ,Model n) M =>
((extChartAt 𝓘(ℝ,Model n) a) z.1,
((extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2),
(extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))
(⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) z).2))) ''
{z : TangentBundle 𝓘(ℝ,Model n) M |
z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L}
lemma isCompact_minimizingRays_between {K L : Set M}
(hK : IsCompact K) (hL : IsCompact L) :
IsCompact {z : TangentBundle 𝓘(ℝ,Model n) M |
z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L} := by
exact (isCompact_total_minimizingVectors (n := n) (M := M)).inter_right
((hK.isClosed.preimage (FiberBundle.continuous_proj _ _)).inter
(hL.isClosed.preimage contMDiff_riemannianExp.continuous))
lemma continuousOn_riemannianExp_chart (c : M) :
ContinuousOn (fun z : TangentBundle 𝓘(ℝ,Model n) M =>
(extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2))
{z | riemannianExp z.1 z.2∈(extChartAt 𝓘(ℝ,Model n) c).source} :=
(continuousOn_extChartAt c).comp contMDiff_riemannianExp.continuous.continuousOn
(fun _ hz => hz)
lemma continuousOn_endpointChartParameters {a c : M} {K L : Set M}
(hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source)
(hLs : L⊆(extChartAt 𝓘(ℝ,Model n) c).source) :
ContinuousOn (fun z : TangentBundle 𝓘(ℝ,Model n) M =>
((extChartAt 𝓘(ℝ,Model n) a) z.1,
((extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2),
(extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))
(⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) z).2)))
{z : TangentBundle 𝓘(ℝ,Model n) M |
z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L} := by
let S := {z : TangentBundle 𝓘(ℝ,Model n) M |
z.2∈minimizingVectors z.1 ∧ z.1∈K ∧ riemannianExp z.1 z.2∈L}
have hX : ContinuousOn (extChartAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n))
(⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M)) S :=
(continuousOn_extChartAt _).mono
(fun z hz => (tangent_chart_source_iff _ z).mpr (hKs hz.2.1))
have hY : ContinuousOn (fun z : TangentBundle 𝓘(ℝ,Model n) M =>
(extChartAt 𝓘(ℝ,Model n) c) (riemannianExp z.1 z.2)) S :=
(continuousOn_riemannianExp_chart c).mono (fun z hz => hLs hz.2.2)
have H := hX.fst.prodMk (hY.prodMk hX.snd)
convert H using 1
funext z
rw [tangent_chart_apply]
lemma isCompact_minimizingEndpointChartParameters {a c : M} {K L : Set M}
(hK : IsCompact K) (hL : IsCompact L)
(hKs : K⊆(extChartAt 𝓘(ℝ,Model n) a).source)
(hLs : L⊆(extChartAt 𝓘(ℝ,Model n) c).source) :
IsCompact (minimizingEndpointChartParameters (n := n) a c K L) :=
(isCompact_minimizingRays_between hK hL).image_of_continuousOn
(continuousOn_endpointChartParameters hKs hLs)
omit [CompactSpace M]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
omit [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] in
lemma tangent_symmL_trivialization {a x : M}
(hx : x∈(extChartAt 𝓘(ℝ,Model n) a).source)
(p : TangentSpace 𝓘(ℝ,Model n) x) :
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).symmL ℝ x
((trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p)=p :=
Trivialization.symmL_continuousLinearMapAt _
(by simpa only [TangentBundle.trivializationAt_baseSet,extChartAt_source] using hx) p
lemma trivialization_mem_minimizingEndpointChartParameters {a c x : M} {K L : Set M}
(hx : x∈K) (hxa : x∈(extChartAt 𝓘(ℝ,Model n) a).source)
{p : TangentSpace 𝓘(ℝ,Model n) x} (hp : p∈minimizingVectors x)
(hy : riemannianExp x p∈L) :
((extChartAt 𝓘(ℝ,Model n) a) x,
((extChartAt 𝓘(ℝ,Model n) c) (riemannianExp x p),
(trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x p))∈
minimizingEndpointChartParameters a c K L := by
refine ⟨⟨x,p⟩,⟨hp,hx,hy⟩,?_⟩
simp only [tangent_chart_trivialization hxa]
end WeakMTWTransport
end
end OAI
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