Theorems · Theorem · global analysis

isLocalStructomorphOn_contDiffGroupoid_iff

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {n : WithTop ℕ∞} {M' : Type u_5}
  [inst_6 : TopologicalSpace M'] [IsManifold I n M] [inst_8 : ChartedSpace H M'] [IsM' : IsManifold I n M']
  (f : OpenPartialHomeomorph M M'),
  ChartedSpace.LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source ↔
    ContMDiffOn I I n (↑f) f.source ∧ ContMDiffOn I I n (↑f.symm) f.target

Let M and M' be manifolds with the same model-with-corners, I. Then f : M → M' is a local structomorphism for I, if and only if it is manifold-C^n on the domain of definition in both directions.

Defined in

Mathlib.Geometry.Manifold.ContMDiff.Atlas

Cited by

0 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceIsManifoldChartedSpaceIsManifold