-
Notifications
You must be signed in to change notification settings - Fork 7
Expand file tree
/
Copy pathMetricInnerSmoothness.lean
More file actions
232 lines (195 loc) · 9.32 KB
/
Copy pathMetricInnerSmoothness.lean
File metadata and controls
232 lines (195 loc) · 9.32 KB
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
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
import OpenGALib.Riemannian.Manifold.SmoothManifold
import OpenGALib.Riemannian.TangentBundle.TangentSmooth
/-!
# `metricInner` smoothness — pointwise / set / global variants
Engineering parity API expanding the Math headlines
* `Riemannian.RiemannianMetric.metricInner_contMDiffWithinAt`
(Metric.lean — real proof via `ContMDiffWithinAt.inner_bundle`)
* `Riemannian.metricInner_contMDiffWithinAt`
(Manifold.lean — typeclass-bound restatement)
into the full `_within_at / _at / _on / _` parity for both the
explicit-metric form `g.metricInner_X` and the typeclass-bound form
`HasMetric.metric.metricInner_X` keyed on `[HasMetric I M]`. Each variant is a pointwise
reduction to the Math headline.
-/
open Bundle
open scoped ContDiff Manifold Bundle
namespace Riemannian.RiemannianMetric
section Smoothness
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
{M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]
{v w : ∀ x : M, TangentSpace I x} {s : Set M} {x : M}
/-! ### `ContMDiff` family — smoothness order `n ≤ ∞` -/
variable {n : ℕ∞ω} [hLE : ENat.LEInfty n]
/-- **Eng.** Pointwise variant. -/
theorem metricInner_contMDiffAt
(g : RiemannianMetric I M)
(hv : ContMDiffAt I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, v y⟩ : TangentBundle I M)) x)
(hw : ContMDiffAt I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, w y⟩ : TangentBundle I M)) x) :
ContMDiffAt I 𝓘(ℝ, ℝ) n
(fun y => g.metricInner y (v y) (w y)) x :=
g.metricInner_contMDiffWithinAt hv hw
/-- **Eng.** Set-form variant. -/
theorem metricInner_contMDiffOn
(g : RiemannianMetric I M)
(hv : ContMDiffOn I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, v y⟩ : TangentBundle I M)) s)
(hw : ContMDiffOn I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, w y⟩ : TangentBundle I M)) s) :
ContMDiffOn I 𝓘(ℝ, ℝ) n
(fun y => g.metricInner y (v y) (w y)) s :=
fun y hy => g.metricInner_contMDiffWithinAt (hv y hy) (hw y hy)
/-- **Eng.** Global variant. -/
theorem metricInner_contMDiff
(g : RiemannianMetric I M)
(hv : ContMDiff I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, v y⟩ : TangentBundle I M)))
(hw : ContMDiff I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, w y⟩ : TangentBundle I M))) :
ContMDiff I 𝓘(ℝ, ℝ) n
(fun y => g.metricInner y (v y) (w y)) :=
fun y => g.metricInner_contMDiffAt (hv y) (hw y)
/-! ### `MDifferentiable` family — first-order differentiability -/
/-- **Eng.** Differentiable-within-at variant. -/
theorem metricInner_mdifferentiableWithinAt
(g : RiemannianMetric I M)
(hv : MDifferentiableWithinAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)) s x)
(hw : MDifferentiableWithinAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M)) s x) :
MDifferentiableWithinAt I 𝓘(ℝ, ℝ)
(fun y => g.metricInner y (v y) (w y)) s x := by
letI rb : Bundle.RiemannianBundle (TangentSpace I : M → Type _) :=
⟨g.toRiemannianMetric⟩
exact MDifferentiableWithinAt.inner_bundle (IB := I) (F := E)
(E := (TangentSpace I : M → Type _)) (b := fun y => y)
(v := v) (w := w) (IM := I) hv hw
/-- **Eng.** Pointwise differentiability. -/
theorem metricInner_mdifferentiableAt
(g : RiemannianMetric I M)
(hv : MDifferentiableAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)) x)
(hw : MDifferentiableAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M)) x) :
MDifferentiableAt I 𝓘(ℝ, ℝ)
(fun y => g.metricInner y (v y) (w y)) x :=
g.metricInner_mdifferentiableWithinAt hv hw
/-- **Eng.** Set-form differentiability. -/
theorem metricInner_mdifferentiableOn
(g : RiemannianMetric I M)
(hv : MDifferentiableOn I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)) s)
(hw : MDifferentiableOn I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M)) s) :
MDifferentiableOn I 𝓘(ℝ, ℝ)
(fun y => g.metricInner y (v y) (w y)) s :=
fun y hy => g.metricInner_mdifferentiableWithinAt (hv y hy) (hw y hy)
/-- **Eng.** Global differentiability. -/
theorem metricInner_mdifferentiable
(g : RiemannianMetric I M)
(hv : MDifferentiable I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)))
(hw : MDifferentiable I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M))) :
MDifferentiable I 𝓘(ℝ, ℝ)
(fun y => g.metricInner y (v y) (w y)) :=
fun y => g.metricInner_mdifferentiableAt (hv y) (hw y)
/-- **Eng.** `TangentSmoothAt`-form pointwise differentiability — convenience
wrapper that converts the framework's `TangentSmoothAt` predicate to
the underlying `MDifferentiableAt` bundle-section form. Explicit-`g` variant. -/
theorem metricInner_mdifferentiableAt_of_tangentSmoothAt
(g : RiemannianMetric I M)
{Y Z : ∀ y : M, TangentSpace I y}
(hY : TangentSmoothAt Y x) (hZ : TangentSmoothAt Z x) :
MDifferentiableAt I 𝓘(ℝ, ℝ)
(fun y => g.metricInner y (Y y) (Z y)) x :=
g.metricInner_mdifferentiableAt hY.toBundleSection hZ.toBundleSection
end Smoothness
end Riemannian.RiemannianMetric
namespace Riemannian
section MetricInnerSmoothness
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
{M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M]
[hm : HasMetric I M]
{v w : ∀ x : M, TangentSpace I x} {s : Set M} {x : M}
/-! ### `ContMDiff` family — smoothness order `n ≤ ∞` -/
variable {n : ℕ∞ω} [hLE : ENat.LEInfty n]
/-- **Eng.** Pointwise variant. -/
theorem metricInner_contMDiffAt
(hv : ContMDiffAt I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, v y⟩ : TangentBundle I M)) x)
(hw : ContMDiffAt I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, w y⟩ : TangentBundle I M)) x) :
ContMDiffAt I 𝓘(ℝ, ℝ) n
(fun y => HasMetric.metric.metricInner y (v y) (w y)) x :=
hm.metric.metricInner_contMDiffAt hv hw
/-- **Eng.** Set-form variant. -/
theorem metricInner_contMDiffOn
(hv : ContMDiffOn I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, v y⟩ : TangentBundle I M)) s)
(hw : ContMDiffOn I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, w y⟩ : TangentBundle I M)) s) :
ContMDiffOn I 𝓘(ℝ, ℝ) n
(fun y => HasMetric.metric.metricInner y (v y) (w y)) s :=
hm.metric.metricInner_contMDiffOn hv hw
/-- **Eng.** Global variant. -/
theorem metricInner_contMDiff
(hv : ContMDiff I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, v y⟩ : TangentBundle I M)))
(hw : ContMDiff I (I.prod 𝓘(ℝ, E)) n
(fun y => (⟨y, w y⟩ : TangentBundle I M))) :
ContMDiff I 𝓘(ℝ, ℝ) n
(fun y => HasMetric.metric.metricInner y (v y) (w y)) :=
hm.metric.metricInner_contMDiff hv hw
/-! ### `MDifferentiable` family — first-order differentiability -/
/-- **Eng.** Differentiable-within-at variant. -/
theorem metricInner_mdifferentiableWithinAt
(hv : MDifferentiableWithinAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)) s x)
(hw : MDifferentiableWithinAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M)) s x) :
MDifferentiableWithinAt I 𝓘(ℝ, ℝ)
(fun y => HasMetric.metric.metricInner y (v y) (w y)) s x :=
hm.metric.metricInner_mdifferentiableWithinAt hv hw
/-- **Eng.** Pointwise differentiability. -/
theorem metricInner_mdifferentiableAt
(hv : MDifferentiableAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)) x)
(hw : MDifferentiableAt I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M)) x) :
MDifferentiableAt I 𝓘(ℝ, ℝ)
(fun y => HasMetric.metric.metricInner y (v y) (w y)) x :=
hm.metric.metricInner_mdifferentiableAt hv hw
/-- **Eng.** Set-form differentiability. -/
theorem metricInner_mdifferentiableOn
(hv : MDifferentiableOn I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)) s)
(hw : MDifferentiableOn I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M)) s) :
MDifferentiableOn I 𝓘(ℝ, ℝ)
(fun y => HasMetric.metric.metricInner y (v y) (w y)) s :=
hm.metric.metricInner_mdifferentiableOn hv hw
/-- **Eng.** Global differentiability. -/
theorem metricInner_mdifferentiable
(hv : MDifferentiable I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, v y⟩ : TangentBundle I M)))
(hw : MDifferentiable I (I.prod 𝓘(ℝ, E))
(fun y => (⟨y, w y⟩ : TangentBundle I M))) :
MDifferentiable I 𝓘(ℝ, ℝ)
(fun y => HasMetric.metric.metricInner y (v y) (w y)) :=
hm.metric.metricInner_mdifferentiable hv hw
/-- **Eng.** `TangentSmoothAt`-form pointwise differentiability — convenience
wrapper that converts the framework's `TangentSmoothAt` predicate to
the underlying `MDifferentiableAt` bundle-section form. -/
theorem metricInner_mdifferentiableAt_of_tangentSmoothAt
{Y Z : ∀ y : M, TangentSpace I y} {x : M}
(hY : TangentSmoothAt Y x) (hZ : TangentSmoothAt Z x) :
MDifferentiableAt I 𝓘(ℝ, ℝ)
(fun y => HasMetric.metric.metricInner y (Y y) (Z y)) x :=
HasMetric.metric.metricInner_mdifferentiableAt hY.toBundleSection hZ.toBundleSection
end MetricInnerSmoothness
end Riemannian