Theorems · Definition · global analysis

Manifold.IsSubmersionAt

{𝕜 : Type u_1} →
  {E'' : Type u_3} →
    {H : Type u_7} →
      {G : Type u_9} →
        {E : Type u} →
          [inst : NontriviallyNormedField 𝕜] →
            [inst_1 : NormedAddCommGroup E] →
              [inst_2 : NormedSpace 𝕜 E] →
                [inst_3 : NormedAddCommGroup E''] →
                  [inst_4 : NormedSpace 𝕜 E''] →
                    [inst_5 : TopologicalSpace H] →
                      [inst_6 : TopologicalSpace G] →
                        ModelWithCorners 𝕜 E H →
                          ModelWithCorners 𝕜 E'' G →
                            {M : Type u_11} →
                              {N : Type u_13} →
                                [inst : TopologicalSpace M] →
                                  [ChartedSpace H M] →
                                    [inst : TopologicalSpace N] → [ChartedSpace G N] → WithTop ℕ∞ → (M → N) → M → Prop

f : M → N is a C^n submersion 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, v) ↦ u. Additionally, we demand that f map φ.source into ψ.source. NB. We don't know the particular atlasses 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. Implicit in this definition is an abstract choice F of a complement of E'' in E: being a submersion at x includes a choice of linear isomorphism between E and E'' × F, which is where the choice of F enters. If you need stronger control over the complement F, use IsSubmersionAtOfComplement instead.

Defined in

Mathlib.Geometry.Manifold.Submersion

Cited by

20 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace