Theorems · Theorem · global analysis

ContMDiffWithinAt.mpullback_vectorField_preimage

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {H' : Type u_5} [inst_6 : TopologicalSpace H']
  {E' : Type u_6} [inst_7 : NormedAddCommGroup E'] [inst_8 : NormedSpace 𝕜 E'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {f : M → M'} {x₀ : M}
  {V : (x : M') → TangentSpace I' x} {m n : WithTop ℕ∞} {t : Set M'} [CompleteSpace E] [inst_12 : IsManifold I 1 M]
  [inst_13 : IsManifold I' 1 M'],
  ContMDiffWithinAt I' I'.tangent m (fun x => ⟨x, V x⟩) t (f x₀) →
    ContMDiffAt I I' n f x₀ →
      (mfderiv% f x₀).IsInvertible →
        m + 1 ≤ n → ContMDiffWithinAt I I.tangent m (fun x => ⟨x, VectorField.mpullback I I' f V x⟩) (f ⁻¹' t) x₀

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 within a set at a point, but with full pullback.

Defined in

Mathlib.Geometry.Manifold.VectorField.Pullback

Cited by

4 results in Mathlib

Foundations

Depth 223 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceCompleteSpaceIsManifoldIsManifold