Theorems · Definition · global analysis

Manifold.IsImmersionAtOfComplement

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E : Type u_2} →
      {E'' : Type u} →
        (F : Type u_5) →
          [inst_1 : NormedAddCommGroup E] →
            [inst_2 : NormedSpace 𝕜 E] →
              [inst_3 : NormedAddCommGroup E''] →
                [inst_4 : NormedSpace 𝕜 E''] →
                  [inst_5 : NormedAddCommGroup F] →
                    [NormedSpace 𝕜 F] →
                      {H : Type u_7} →
                        [inst_7 : TopologicalSpace H] →
                          {G : Type u_9} →
                            [inst_8 : TopologicalSpace G] →
                              ModelWithCorners 𝕜 E H →
                                ModelWithCorners 𝕜 E'' G →
                                  {M : Type u_11} →
                                    [inst : TopologicalSpace M] →
                                      [ChartedSpace H M] →
                                        {N : Type u_13} →
                                          [inst : TopologicalSpace N] →
                                            [ChartedSpace G N] → WithTop ℕ∞ → (M → N) → M → Prop

f : M → N is a C^n immersion at x if there are charts φ and ψ of M and N around x and f x, respectively such that in these charts, f looks like u ↦ (u, 0). Additionally, we demand that f map φ.source into ψ.source. NB. We don't know the particular atlases used for M and N, so asking for φ and ψ to be in the atlas would be too optimistic: lying in the maximalAtlas is sufficient. This definition has a fixed parameter F, which is a choice of complement of E in E': being an immersion at x includes a choice of linear isomorphism between E × F and E'. While the particular choice of complement is often not important, choosing a complement is useful in some settings, such as proving that embedded submanifolds are locally given either by an immersion or a submersion. Unless you have a particular reason, prefer to use IsImmersionAt instead.

Defined in

Mathlib.Geometry.Manifold.Immersion

Cited by

36 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace