Theorems · Definition · global analysis

Manifold.LiftSourceTargetPropertyAt

{𝕜 : Type u_1} →
  {E : Type u_2} →
    {F : Type u_4} →
      {H : Type u_6} →
        {G : Type u_8} →
          [inst : NontriviallyNormedField 𝕜] →
            [inst_1 : NormedAddCommGroup E] →
              [inst_2 : NormedSpace 𝕜 E] →
                [inst_3 : NormedAddCommGroup F] →
                  [inst_4 : NormedSpace 𝕜 F] →
                    [inst_5 : TopologicalSpace H] →
                      [inst_6 : TopologicalSpace G] →
                        ModelWithCorners 𝕜 E H →
                          ModelWithCorners 𝕜 F G →
                            {M : Type u_10} →
                              {N : Type u_12} →
                                [inst : TopologicalSpace M] →
                                  [ChartedSpace H M] →
                                    [inst_8 : TopologicalSpace N] →
                                      [ChartedSpace G N] →
                                        WithTop ℕ∞ →
                                          (M → N) →
                                            M →
                                              ((M → N) → OpenPartialHomeomorph M H → OpenPartialHomeomorph N G → Prop) →
                                                Prop

The induced property by a local property P: it is satisfied for f at x iff there exist charts φ and ψ of M and N around x and f x, respectively, such that f satisfies P w.r.t. φ and ψ. The motivating examples are smooth immersions and submersions: the corresponding condition is that f look like the inclusion u ↦ (u, 0) (resp. a projection (u, v) ↦ u) in the charts φ and ψ.

Defined in

Mathlib.Geometry.Manifold.LocalSourceTargetProperty

Cited by

17 results in Mathlib

Foundations

Depth 13 from the axioms · uses no axioms

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace