feat(geometry/manifold): Integral curve of vector fields

src/geometry/manifold/integral_curve.lean

@@ -0,0 +1,290 @@
+/-
+Copyright (c) 2022 Winston Yin. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Winston Yin
+-/
+import analysis.ODE.picard_lindelof
+import geometry.manifold.cont_mdiff
+import geometry.manifold.mfderiv
+
+/-!
+# Integral curves of vector fields on a manifold
+
+For any continuously differentiable vector field on a manifold `M` and any chosen non-boundary point
+`x₀ : M`, an integral curve `γ : ℝ → M` exists such that `γ 0 = x₀` and the tangent vector of `γ` at
+`t` coincides with the vector field at `γ t` for all `t` within an open interval around 0.
+
+As a corollary, such an integral curve exists for any starting point `x₀` if `M` is a manifold
+without boundary.
+
+## Implementation details
+
+Since there is already an ODE solution existence theorem
+`ODE_solution_exists.at_ball_of_cont_diff_on_nhds`, the bulk of this file is to convert statements
+about manifolds to statements about the model space. This comes in a few steps:
+1. Express the smoothness of the vector field `v` in a single fixed chart around the starting point
+`x₀`.
+2. Use the ODE solution existence theorem to obtain a curve `γ : ℝ → M` whose derivative coincides
+with the vector field (stated in the local chart around `x₀`).
+3. Same as 2 but now stated in the local chart around `γ t`, which is how `cont_mdiff` is defined.
+
+## Tags
+
+integral curve, vector field
+-/
+
+localized "notation (name := ext_chart_at) `𝓔(` I `, ` x `)` :=
+  ext_chart_at I x" in manifold
+
+open_locale manifold
+
+namespace model_with_corners
+
+variables
+  {𝕜 : Type*} [nontrivially_normed_field 𝕜]
+  {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
+  {H : Type*} [topological_space H] (I : model_with_corners 𝕜 E H)
+  {M : Type*} [topological_space M] [charted_space H M]
+
+lemma boundaryless.is_open_target
+  [I.boundaryless] {x : M} : is_open (ext_chart_at I x).target :=
+begin
+  rw [ext_chart_at_target, model_with_corners.boundaryless.range_eq_univ, set.inter_univ],
+  exact (model_with_corners.continuous_symm _).is_open_preimage _ (local_homeomorph.open_target _)
+end
+
+/-- An interior point of a manifold is a point whose image in the model vector space is in the
+interior of the chart's target.
+
+TODO : This definition refers to a chosen chart at `x`, but the property is independent of the
+choice of chart. See Theorem 1.37 (for general manifolds) and Theorem 1.46 (for smooth manifolds) of
+Introduction to Smooth Manifolds by John M. Lee.
+-/
+def is_interior_point (x : M) := 𝓔(I, x) x ∈ interior 𝓔(I, x).target
+
+lemma boundaryless.is_interior_point
+  [I.boundaryless] {x : M} : I.is_interior_point x :=
+begin
+  rw [is_interior_point, is_open.interior_eq (boundaryless.is_open_target I)],
+  exact local_equiv.map_source _ (mem_ext_chart_source _ _)
+end
+
+end model_with_corners
+
+section
+
+variables
+  {𝕜 : Type*} [nontrivially_normed_field 𝕜]
+  {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]
+  {H : Type*} [topological_space H] {I : model_with_corners 𝕜 E H}
+  {M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
+  {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E']
+  {H' : Type*} [topological_space H'] {I' : model_with_corners 𝕜 E' H'}
+  {M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']
+
+/-- Express cont_mdiff_at in a fixed chosen local chart.
+
+TODO: cont_mdiff_within_at, cont_mdiff_on versions -/
+lemma cont_mdiff_at_indep_ext_chart
+  {n : ℕ∞} {f : M → M'} (x₀ : M) {x : M}
+  (hx : x ∈ 𝓔(I, x₀).source) (hfx : f x ∈ 𝓔(I', f x₀).source) :
+  cont_mdiff_at I I' n f x ↔ continuous_at f x ∧
+    cont_diff_within_at 𝕜 n (written_in_ext_chart_at I I' x₀ f) (set.range I) (𝓔(I, x₀) x) :=
+begin
+  rw [cont_mdiff_at, cont_mdiff_within_at],
+  rw ext_chart_at_source at hx hfx,
+  rw (cont_diff_within_at_local_invariant_prop I I' n).lift_prop_within_at_indep_chart
+    (smooth_manifold_with_corners.chart_mem_maximal_atlas I x₀) hx
+    (smooth_manifold_with_corners.chart_mem_maximal_atlas I' (f x₀)) hfx,
+  apply and_congr (continuous_within_at_iff_continuous_at
+    (is_open.mem_nhds is_open_univ (set.mem_univ x))),
+  rw [cont_diff_within_at_prop, written_in_ext_chart_at, ext_chart_at_coe, ext_chart_at_coe,
+    ext_chart_at_coe_symm, set.inter_comm _ (set.range I)],
+  refine cont_diff_within_at_inter ((I.continuous_at_symm).preimage_mem_nhds _),
+  rw I.left_inv,
+  apply (local_homeomorph.continuous_at_symm _
+    (local_homeomorph.map_source _ hx)).preimage_mem_nhds,
+  exact is_open.mem_nhds is_open_univ (set.mem_univ _)
+end
+
+lemma vector_field_cont_mdiff_at_indep_ext_chart
+  {n : ℕ∞} {v : M → tangent_bundle I M} (hv : ∀ x, (v x).1 = x) (x₀ : M) {x : M}
+  (hx : x ∈ 𝓔(I, x₀).source) :
+  cont_mdiff_at I I.tangent n v x ↔ continuous_at v x ∧
+    cont_diff_within_at 𝕜 n (written_in_ext_chart_at I I.tangent x₀ v) (set.range I) (𝓔(I, x₀) x) :=
+begin
+  refine cont_mdiff_at_indep_ext_chart x₀ hx _,
+  rw [ext_chart_at_source, basic_smooth_vector_bundle_core.mem_chart_source_iff, hv, hv,
+    ←ext_chart_at_source I],
+  exact hx
+end
+
+/-- If `v` is a smooth vector field, then the partial function `E → E` defined by its trivialisation
+on a chosen local chart is smooth. -/
+lemma vector_field_cont_diff_on_snd_of_cont_mdiff
+  {n : ℕ∞} {v : M → tangent_bundle I M} (h₁ : ∀ x, (v x).1 = x)
+  (h₂ : cont_mdiff I I.tangent n v) (x₀ : M) :
+  cont_diff_on 𝕜 n (λ (y : E), (written_in_ext_chart_at I I.tangent x₀ v y).2) 𝓔(I, x₀).target :=
+begin
+  intros y hy,
+  apply cont_diff_within_at.mono _ (ext_chart_at_target_subset_range _ _),
+  rw ←local_equiv.right_inv _ hy,
+  refine cont_diff_at.comp_cont_diff_within_at _ cont_diff_at_snd _,
+  apply ((vector_field_cont_mdiff_at_indep_ext_chart h₁ _ _).mp h₂.cont_mdiff_at).2,
+  exact local_equiv.map_target _ hy
+end
+
+/-- Express the change of coordinates in the tangent bundle in terms of the change of
+  coordinates in the base space. -/
+lemma tangent_bundle_core_coord_change_triv
+  (v v' : tangent_bundle I M) :
+  (𝓔(I.tangent, v') v).2 =
+    (fderiv_within 𝕜 (𝓔(I, v'.1) ∘ 𝓔(I, v.1).symm) (set.range I) (𝓔(I, v.1) v.1)) v.2 := rfl
+
+lemma tangent_bundle_core_coord_change_triv'
+  (v v' : tangent_bundle I M) (hv : v.1 ∈ 𝓔(I, v'.1).source) :
+  (𝓔(I.tangent, v) v).2 =
+    (fderiv_within 𝕜 (𝓔(I, v.1) ∘ 𝓔(I, v'.1).symm) (set.range I) (𝓔(I, v'.1) v.1))
+      (𝓔(I.tangent, v') v).2 :=
+begin
+  rw [ext_chart_at_coe, function.comp_apply, model_with_corners.prod_apply,
+    basic_smooth_vector_bundle_core.to_charted_space_chart_at,
+    basic_smooth_vector_bundle_core.chart_apply],
+  have hi := mem_achart_source H v.1,
+  have hj : v.1 ∈ (achart H v'.1).val.to_local_equiv.source,
+  { rw ext_chart_at_source at hv,
+    exact hv },
+  rw ←basic_smooth_vector_bundle_core.coord_change_comp' _ hi hj hi,
+  refl
+end
+
+end
+
+variables
+  {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
+  {H : Type*} [topological_space H] (I : model_with_corners ℝ E H)
+  (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]
+  (v : M → tangent_bundle I M)
+
+/-- We apply the ODE existence theorem to a continuously differentiable vector field written in the
+  preferred chart around the base point. We require that the base point not be on the boundary.
+  Several useful properties of the solution are proven here, to be used in
+  `exists_integral_curve_of_cont_mdiff_tangent_vector_field`. -/
+lemma exists_integral_curve_of_cont_mdiff_tangent_vector_field_aux [proper_space E]
+  (hv : ∀ x, (v x).1 = x) (hcd : cont_mdiff I I.tangent 1 v) (t₀ : ℝ) (x₀ : M)
+  (hx : I.is_interior_point x₀) :
+  ∃ (ε > (0 : ℝ)) (γ : ℝ → M), γ t₀ = x₀ ∧ ∀ (t : ℝ), t ∈ set.Ioo (t₀ - ε) (t₀ + ε) →
+    (γ t) ∈ 𝓔(I, x₀).source ∧
+    𝓔(I, x₀) (γ t) ∈ interior 𝓔(I, x₀).target ∧
+    continuous_at γ t ∧
+    has_deriv_at (𝓔(I, x₀) ∘ γ) (𝓔(I.tangent, v x₀) (v (γ t))).2 t :=
+begin
+  have hx1 := (vector_field_cont_diff_on_snd_of_cont_mdiff hv hcd x₀).mono interior_subset,
+  have hx2 := is_open.mem_nhds (is_open_interior) hx,
+  obtain ⟨ε, hε, f, hf1, hf2⟩ :=
+    exists_forall_deriv_at_Ioo_eq_of_cont_diff_on_nhds t₀ (𝓔(I, x₀) x₀) hx1 hx2,
+  have hf1' : (𝓔(I, x₀).symm ∘ f) t₀ = x₀,
+  { rw function.comp_apply,
+    rw hf1,
+    exact ext_chart_at_to_inv I x₀ },
+  refine ⟨ε, hε, 𝓔(I, x₀).symm ∘ f, hf1', _⟩,
+  intros t ht,
+  obtain ⟨hf3, hf4⟩ := hf2 t ht,
+  refine ⟨_, _, _, _⟩,
+  { rw [function.comp_apply, ←set.mem_preimage],
+    apply set.mem_of_mem_of_subset _ (local_equiv.target_subset_preimage_source _),
+    apply set.mem_of_mem_of_subset _
+      (interior_subset : interior 𝓔(I, x₀).target ⊆ 𝓔(I, x₀).target),
+    rw ←set.mem_preimage,
+    exact hf3 },
+  { rw [function.comp_apply, ←set.mem_preimage, ←set.mem_preimage],
+    apply set.mem_of_mem_of_subset _ (set.inter_subset_right 𝓔(I, x₀).target _),
+    rw [local_equiv.target_inter_inv_preimage_preimage,
+      set.inter_eq_self_of_subset_right interior_subset],
+    exact hf3 },
+  { refine continuous_at.comp _ hf4.continuous_at,
+    apply ext_chart_continuous_at_symm'',
+    exact set.mem_of_mem_of_subset hf3 interior_subset },
+  { rw [function.comp_apply, ←function.comp_apply v,
+    ←function.comp_apply 𝓔(I.tangent, v x₀), ←written_in_ext_chart_at],
+    apply has_deriv_at.congr_of_eventually_eq hf4,
+    rw filter.eventually_eq_iff_exists_mem,
+    refine ⟨set.Ioo (t₀ - ε) (t₀ + ε), is_open.mem_nhds is_open_Ioo ht, _⟩,
+    intros t' ht',
+    rw [function.comp_apply, function.comp_apply],
+    apply local_equiv.right_inv,
+    exact set.mem_of_mem_of_subset (hf2 t' ht').1 interior_subset }
+end
+
+-- how to generalise / simplify?
+/-- The derivative of a curve on a manifold is independent of the chosen extended chart.
+
+Note: The hypothesis `hγ₂` is equivalent to `I.is_boundary_point (γ t)`, but the equivalence has
+not been implemented in mathlib yet. -/
+lemma curve_has_deriv_at_coord_change
+  (hv : ∀ x, (v x).1 = x) (x₀ : M) (γ : ℝ → M) (t : ℝ)
+  (hγ₁ : (γ t) ∈ 𝓔(I, x₀).source) (hγ₂ : 𝓔(I, x₀) (γ t) ∈ interior 𝓔(I, x₀).target)
+  (hd : has_deriv_at (𝓔(I, x₀) ∘ γ) (𝓔(I.tangent, v x₀) (v (γ t))).2 t) :
+  has_deriv_at ((𝓔(I, γ t) ∘ 𝓔(I, x₀).symm) ∘ (𝓔(I, x₀) ∘ γ))
+    (𝓔(I.tangent, v (γ t)) (v (γ t))).2 t :=
+begin
+  have : (v (γ t)).fst ∈ 𝓔(I, (v x₀).1).source,
+  { rw [hv, hv],
+    exact hγ₁ },
+  rw tangent_bundle_core_coord_change_triv' (v (γ t)) (v x₀) this,
+  apply has_fderiv_at.comp_has_deriv_at _ _ hd,
+  rw [hv, hv, function.comp_apply],
+  have : set.range I ∈ nhds (𝓔(I, x₀) (γ t)),
+  { rw mem_nhds_iff,
+    refine ⟨interior 𝓔(I, x₀).target, _, is_open_interior, hγ₂⟩,
+    refine set.subset.trans interior_subset _,
+    rw ext_chart_at_target,
+    exact set.inter_subset_right _ _ },
+  apply has_fderiv_within_at.has_fderiv_at _ this,
+  apply differentiable_within_at.has_fderiv_within_at,
+  apply cont_diff_within_at.differentiable_within_at _ le_top,
+  apply cont_diff_within_at_ext_coord_change,
+  apply local_equiv.mem_symm_trans_source _ hγ₁,
+  exact mem_ext_chart_source _ _
+end
+
+/-- For any continuously differentiable vector field and any chosen non-boundary point `x₀` on the
+  manifold, an integral curve `γ : ℝ → M` exists such that `γ 0 = x₀` and the tangent vector of `γ`
+  at `t` coincides with the vector field at `γ t` for all `t` within an open interval around 0.-/
+theorem exists_integral_curve_of_cont_mdiff_tangent_vector_field [proper_space E]
+  (hv : ∀ x, (v x).1 = x) (hcd : cont_mdiff I I.tangent 1 v)
+  (t₀ : ℝ) (x₀ : M) (hx : I.is_interior_point x₀) :
+  ∃ (ε > (0 : ℝ)) (γ : ℝ → M), γ t₀ = x₀ ∧ ∀ (t : ℝ), t ∈ set.Ioo (t₀ - ε) (t₀ + ε) →
+    has_mfderiv_at 𝓘(ℝ, ℝ) I γ t ((1 : ℝ →L[ℝ] ℝ).smul_right (𝓔(I.tangent, v(γ t)) (v (γ t))).2) :=
+begin
+  obtain ⟨ε, hε, γ, hf1, hf2⟩ :=
+    exists_integral_curve_of_cont_mdiff_tangent_vector_field_aux I M v hv hcd t₀ x₀ hx,
+  refine ⟨ε, hε, γ, hf1, _⟩,
+  intros t ht,
+  rw has_mfderiv_at,
+  obtain ⟨hf3, hf4, hf5, hf6⟩ := hf2 t ht,
+  use hf5,
+  rw [ext_chart_model_space_apply, written_in_ext_chart_at, ext_chart_model_space_eq_id,
+    local_equiv.refl_symm, local_equiv.refl_coe, function.comp.right_id],
+  apply has_deriv_within_at.has_fderiv_within_at,
+  apply has_deriv_at.has_deriv_within_at,
+  have hd := curve_has_deriv_at_coord_change I M v hv x₀ γ t hf3 hf4 hf6,
+  apply has_deriv_at.congr_of_eventually_eq hd,
+  rw filter.eventually_eq_iff_exists_mem,
+  refine ⟨set.Ioo (t₀ - ε) (t₀ + ε), is_open.mem_nhds is_open_Ioo ht, _⟩,
+  intros t' ht',
+  rw [function.comp_apply, function.comp_apply, function.comp_apply, local_equiv.left_inv],
+  exact (hf2 t' ht').1
+end
+
+/-- For any continuously differentiable vector field defined on a manifold without boundary and any
+  chosen starting point `x₀ : M`, an integral curve `γ : ℝ → M` exists such that `γ 0 = x₀` and the
+  tangent vector of `γ` at `t` coincides with the vector field at `γ t` for all `t` within an open
+  interval around 0. -/
+lemma exists_integral_curve_of_cont_mdiff_tangent_vector_field_of_boundaryless
+  [proper_space E] [I.boundaryless]
+  (h₁ : ∀ x, (v x).1 = x) (h₂ : cont_mdiff I I.tangent 1 v) (t₀ : ℝ) (x₀ : M) :
+  ∃ (ε > (0 : ℝ)) (γ : ℝ → M), γ t₀ = x₀ ∧ ∀ (t : ℝ), t ∈ set.Ioo (t₀ - ε) (t₀ + ε) →
+    has_mfderiv_at 𝓘(ℝ, ℝ) I γ t ((1 : ℝ →L[ℝ] ℝ).smul_right (𝓔(I.tangent, v(γ t)) (v (γ t))).2) :=
+exists_integral_curve_of_cont_mdiff_tangent_vector_field I M v h₁ h₂ t₀ x₀
+    (model_with_corners.boundaryless.is_interior_point I)