The groupoid of smooth, fiberwise-linear maps

This file contains preliminaries for the definition of a smooth vector bundle: an associated StructureGroupoid, the groupoid of smoothFiberwiseLinear functions.

The groupoid of smooth, fiberwise-linear maps

def FiberwiseLinear.partialHomeomorph {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {U : Set B} (φ : B → F ≃L[𝕜] F) (hU : IsOpen U) (hφ : ContinuousOn (fun (x : B) => ↑(φ x)) U) (h2φ : ContinuousOn (fun (x : B) => ↑(φ x).symm) U) : PartialHomeomorph (B × F) (B × F)

For B a topological space and F a 𝕜-normed space, a map from U : Set B to F ≃L[𝕜] F determines a partial homeomorphism from B × F to itself by its action fiberwise.

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theorem FiberwiseLinear.trans_partialHomeomorph_apply {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ φ' : B → F ≃L[𝕜] F} {U U' : Set B} (hU : IsOpen U) (hφ : ContinuousOn (fun (x : B) => ↑(φ x)) U) (h2φ : ContinuousOn (fun (x : B) => ↑(φ x).symm) U) (hU' : IsOpen U') (hφ' : ContinuousOn (fun (x : B) => ↑(φ' x)) U') (h2φ' : ContinuousOn (fun (x : B) => ↑(φ' x).symm) U') (b : B) (v : F) : ↑((FiberwiseLinear.partialHomeomorph φ hU hφ h2φ).trans (FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ')) (b, v) = (b, (φ' b) ((φ b) v))

Compute the composition of two partial homeomorphisms induced by fiberwise linear equivalences.

theorem FiberwiseLinear.source_trans_partialHomeomorph {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ φ' : B → F ≃L[𝕜] F} {U U' : Set B} (hU : IsOpen U) (hφ : ContinuousOn (fun (x : B) => ↑(φ x)) U) (h2φ : ContinuousOn (fun (x : B) => ↑(φ x).symm) U) (hU' : IsOpen U') (hφ' : ContinuousOn (fun (x : B) => ↑(φ' x)) U') (h2φ' : ContinuousOn (fun (x : B) => ↑(φ' x).symm) U') : ((FiberwiseLinear.partialHomeomorph φ hU hφ h2φ).trans (FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ')).source = (U ∩ U') ×ˢ Set.univ

Compute the source of the composition of two partial homeomorphisms induced by fiberwise linear equivalences.

theorem FiberwiseLinear.target_trans_partialHomeomorph {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ φ' : B → F ≃L[𝕜] F} {U U' : Set B} (hU : IsOpen U) (hφ : ContinuousOn (fun (x : B) => ↑(φ x)) U) (h2φ : ContinuousOn (fun (x : B) => ↑(φ x).symm) U) (hU' : IsOpen U') (hφ' : ContinuousOn (fun (x : B) => ↑(φ' x)) U') (h2φ' : ContinuousOn (fun (x : B) => ↑(φ' x).symm) U') : ((FiberwiseLinear.partialHomeomorph φ hU hφ h2φ).trans (FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ')).target = (U ∩ U') ×ˢ Set.univ

Compute the target of the composition of two partial homeomorphisms induced by fiberwise linear equivalences.

theorem SmoothFiberwiseLinear.locality_aux₁ {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] {IB : ModelWithCorners 𝕜 EB HB} (e : PartialHomeomorph (B × F) (B × F)) (h : ∀ p ∈ e.source, ∃ (s : Set (B × F)), IsOpen s ∧ p ∈ s ∧ ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (hφ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x)) u) (h2φ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x).symm) u), (e.restr s).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)) : ∃ (U : Set B), e.source = U ×ˢ Set.univ ∧ ∀ x ∈ U, ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x)) u) (h2φ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x).symm) u), (e.restr (u ×ˢ Set.univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)

Let e be a partial homeomorphism of B × F. Suppose that at every point p in the source of e, there is some neighbourhood s of p on which e is equal to a bi-smooth fiberwise linear partial homeomorphism. Then the source of e is of the form U ×ˢ univ, for some set U in B, and, at any point x in U, admits a neighbourhood u of x such that e is equal on u ×ˢ univ to some bi-smooth fiberwise linear partial homeomorphism.

theorem SmoothFiberwiseLinear.locality_aux₂ {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] {IB : ModelWithCorners 𝕜 EB HB} (e : PartialHomeomorph (B × F) (B × F)) (U : Set B) (hU : e.source = U ×ˢ Set.univ) (h : ∀ x ∈ U, ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x)) u) (h2φ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x).symm) u), (e.restr (u ×ˢ Set.univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)) : ∃ (Φ : B → F ≃L[𝕜] F) (U : Set B) (hU₀ : IsOpen U) (hΦ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(Φ x)) U) (h2Φ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(Φ x).symm) U), e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ ⋯ ⋯)

Let e be a partial homeomorphism of B × F whose source is U ×ˢ univ, for some set U in B, and which, at any point x in U, admits a neighbourhood u of x such that e is equal on u ×ˢ univ to some bi-smooth fiberwise linear partial homeomorphism. Then e itself is equal to some bi-smooth fiberwise linear partial homeomorphism.

This is the key mathematical point of the locality condition in the construction of the StructureGroupoid of bi-smooth fiberwise linear partial homeomorphisms. The proof is by gluing together the various bi-smooth fiberwise linear partial homeomorphism which exist locally.

The U in the conclusion is the same U as in the hypothesis. We state it like this, because this is exactly what we need for smoothFiberwiseLinear.

def smoothFiberwiseLinear {𝕜 : Type u_1} (B : Type u_2) (F : Type u_3) [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] (IB : ModelWithCorners 𝕜 EB HB) : StructureGroupoid (B × F)

For B a manifold and F a normed space, the groupoid on B × F consisting of local homeomorphisms which are bi-smooth and fiberwise linear, and induce the identity on B. When a (topological) vector bundle is smooth, then the composition of charts associated to the vector bundle belong to this groupoid.

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@[simp]

theorem mem_smoothFiberwiseLinear_iff {𝕜 : Type u_1} (B : Type u_2) (F : Type u_3) [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] (IB : ModelWithCorners 𝕜 EB HB) (e : PartialHomeomorph (B × F) (B × F)) : e ∈ smoothFiberwiseLinear B F IB ↔ ∃ (φ : B → F ≃L[𝕜] F) (U : Set B) (hU : IsOpen U) (hφ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x)) U) (h2φ : ContMDiffOn IB (modelWithCornersSelf 𝕜 (F →L[𝕜] F)) ⊤ (fun (x : B) => ↑(φ x).symm) U), e.EqOnSource (FiberwiseLinear.partialHomeomorph φ hU ⋯ ⋯)