Theorems · Definition · global analysis

Diffeomorph.smul

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {H : Type u_2} →
      [inst_1 : TopologicalSpace H] →
        {E : Type u_3} →
          [inst_2 : NormedAddCommGroup E] →
            [inst_3 : NormedSpace 𝕜 E] →
              (I : ModelWithCorners 𝕜 E H) →
                {H' : Type u_4} →
                  [inst_4 : TopologicalSpace H'] →
                    {E' : Type u_5} →
                      [inst_5 : NormedAddCommGroup E'] →
                        [inst_6 : NormedSpace 𝕜 E'] →
                          (I' : ModelWithCorners 𝕜 E' H') →
                            {G : Type u_8} →
                              [inst_7 : TopologicalSpace G] →
                                [inst_8 : ChartedSpace H G] →
                                  {M : Type u_9} →
                                    [inst_9 : TopologicalSpace M] →
                                      [inst_10 : ChartedSpace H' M] →
                                        [inst_11 : Group G] →
                                          [inst_12 : MulAction G M] →
                                            (n : WithTop ℕ∞) → [ContMDiffSMul I I' n G M] → G → Diffeomorph I' I' M M n

The diffeomorphism given by scalar multiplication by an element of a group G acting Cⁿ-differentiably on a manifold M is a diffeomorphism from M to itself. Its inverse is scalar multiplication by g⁻¹.

Defined in

Mathlib.Geometry.Manifold.Algebra.SMul

Cited by

4 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceTopologicalSpaceChartedSpaceGroupMulActionContMDiffSMul