Theorems · Definition · global analysis

Manifold.IsSubmersionAt.equiv

{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] →
                        {I : ModelWithCorners 𝕜 E H} →
                          {J : ModelWithCorners 𝕜 E'' G} →
                            {M : Type u_11} →
                              {N : Type u_13} →
                                [inst_7 : TopologicalSpace M] →
                                  [inst_8 : ChartedSpace H M] →
                                    [inst_9 : TopologicalSpace N] →
                                      [inst_10 : ChartedSpace G N] →
                                        {n : WithTop ℕ∞} →
                                          {f : M → N} →
                                            {x : M} →
                                              (h : Manifold.IsSubmersionAt I J n f x) → E ≃L[𝕜] E'' × h.complement

A linear equivalence E ≃L[𝕜] (E'' × F) which belongs to the data of a submersion f at x: the particular equivalence is arbitrary, but this choice matches the witnesses given by h.domChart and h.codChart.

Defined in

Mathlib.Geometry.Manifold.Submersion

Cited by

3 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpace