English
In the ext-chart-at symmetric setting, the pullback mpullbackWithin_vectorField_inter_eq holds giving C^m differentiability on the target chart range.
Русский
В симметричной схеме расширенного диаграммного графа, пуллбэк mpullbackWithin_vectorField_inter_eq обеспечивает C^m-дифференцируемость на диапазоне целевой карты.
LaTeX
$$$\text{Under symm extChartAt } ContMDiffWithinAt( ... ) \Rightarrow \text{ pullback is } C^m.$$$
Lean4
/-- The pullback of a `C^m` vector field by a `C^n` function with invertible derivative and
with `m + 1 ≤ n` is `C^m`.
Version on a set, but with full pullback -/
protected theorem _root_.ContMDiffOn.mpullback_vectorField_preimage
(hV : ContMDiffOn I' I'.tangent m (fun (y : M') ↦ (V y : TangentBundle I' M')) t) (hf : ContMDiff I I' n f)
(hf' : ∀ x ∈ f ⁻¹' t, (mfderiv I I' f x).IsInvertible) (hmn : m + 1 ≤ n) :
ContMDiffOn I I.tangent m (fun (y : M) ↦ (mpullback I I' f V y : TangentBundle I M)) (f ⁻¹' t) := fun x₀ hx₀ ↦
ContMDiffWithinAt.mpullback_vectorField_preimage (hV _ hx₀) (hf x₀) (hf' _ hx₀) hmn
