Liouville-preserving diffeomorphisms of cotangent bundles are cotangent lifts

The cotangent lift T*h : (x, p) ↦ (h x, p ∘ (dh_x)⁻¹) of a smooth diffeomorphism h : M → N pulls the Liouville form λ_N back to λ_M (Diffeomorph.mpullback_cotangent_liouvilleForm). This file proves the converse: on Hausdorff manifolds without boundary, every smooth diffeomorphism ψ : T*M → T*N with ψ^* λ_N = λ_M is the cotangent lift of a unique smooth diffeomorphism M → N (Diffeomorph.existsUnique_cotangent_eq_of_mpullback_liouvilleForm). Together with the naturality of λ, this characterizes cotangent lifts among diffeomorphisms of cotangent bundles (Diffeomorph.exists_cotangent_eq_iff_mpullback_liouvilleForm).

The statement is about the full cotangent bundle, including the zero section. The argument has two steps.

• Fibres go to fibres. A Liouville-preserving ψ commutes with the positive fibre dilations (Diffeomorph.apply_mk_smul_of_mpullback_liouvilleForm). Letting the dilation factor tend to 0 and using continuity of ψ at the zero section, ψ sends the zero covector over x to the zero covector over the base point of ψ (x, p), for every covector p at x (Diffeomorph.apply_mk_zero_of_mpullback_liouvilleForm). So ψ covers the base map f x = π (ψ (x, 0)), and it maps the zero section onto the zero section. The same holds for ψ⁻¹, so f is a diffeomorphism.
• The covector is determined. Since π_N ∘ ψ = f ∘ π_M, the identity ψ^* λ_N = λ_M evaluated on a tangent vector of T*M over v ∈ T_x M reads (ψ (x, p)) (df_x v) = p v. Every v lifts, since the projection has surjective derivative (Bundle.mfderiv_proj_surjective). Hence ψ (x, p) = (f x, p ∘ (df_x)⁻¹); in particular ψ is linear on fibres.

Main results

• Diffeomorph.apply_mk_zero_of_mpullback_liouvilleForm: a Liouville-preserving diffeomorphism sends the zero covector over the base point of z to the zero covector over the base point of ψ z.
• Diffeomorph.existsUnique_cotangent_eq_of_mpullback_liouvilleForm: a Liouville-preserving diffeomorphism is the cotangent lift of a unique diffeomorphism of the bases.
• Diffeomorph.exists_cotangent_eq_iff_mpullback_liouvilleForm: a diffeomorphism of cotangent bundles is a cotangent lift exactly when it preserves the Liouville forms.

References

• A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001, Lecture 2 (lifts of diffeomorphisms) and its exercises.

theorem Diffeomorph.apply_mk_zero_of_mpullback_liouvilleForm {E : Type u_1} {H : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] {F : Type u_4} {H' : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [TopologicalSpace H'] {J : ModelWithCorners ℝ F H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H' N] [IsManifold J (↑⊤) N] [T2Space N] [J.Boundaryless] (ψ : Diffeomorph (I.prod (modelWithCornersSelf ℝ (E →L[ℝ] ℝ))) (J.prod (modelWithCornersSelf ℝ (F →L[ℝ] ℝ))) (TauCeti.CotangentBundle I M) (TauCeti.CotangentBundle J N) ↑⊤) (hψ : (TauCeti.SmoothForm.mpullback ⇑ψ ⋯) (TauCeti.CotangentBundle.liouvilleForm J N) = TauCeti.CotangentBundle.liouvilleForm I M) (z : TauCeti.CotangentBundle I M) : ψ ⟨z.proj, 0⟩ = ⟨(ψ z).proj, 0⟩

Liouville-preserving diffeomorphisms preserve the zero section and the fibres. Let ψ : T*M → T*N be a diffeomorphism pulling λ_N back to λ_M. For every covector z, ψ sends the zero covector over the base point of z to the zero covector over the base point of ψ z. In particular the base point of ψ z depends only on that of z.

theorem Diffeomorph.existsUnique_cotangent_eq_of_mpullback_liouvilleForm {E : Type u_1} {H : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] {F : Type u_4} {H' : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [TopologicalSpace H'] {J : ModelWithCorners ℝ F H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H' N] [IsManifold J (↑⊤) N] [T2Space N] [J.Boundaryless] [T2Space M] [I.Boundaryless] (ψ : Diffeomorph (I.prod (modelWithCornersSelf ℝ (E →L[ℝ] ℝ))) (J.prod (modelWithCornersSelf ℝ (F →L[ℝ] ℝ))) (TauCeti.CotangentBundle I M) (TauCeti.CotangentBundle J N) ↑⊤) (hψ : (TauCeti.SmoothForm.mpullback ⇑ψ ⋯) (TauCeti.CotangentBundle.liouvilleForm J N) = TauCeti.CotangentBundle.liouvilleForm I M) : ∃! h : Diffeomorph I J M N ↑⊤, h.cotangent = ψ

Liouville-preserving diffeomorphisms of cotangent bundles are cotangent lifts. On Hausdorff manifolds without boundary, a smooth diffeomorphism ψ : T*M → T*N with ψ^* λ_N = λ_M is the cotangent lift of a unique smooth diffeomorphism h : M → N. Since the lift covers h, necessarily h x = π (ψ (x, 0)).

theorem Diffeomorph.exists_cotangent_eq_iff_mpullback_liouvilleForm {E : Type u_1} {H : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] {F : Type u_4} {H' : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [TopologicalSpace H'] {J : ModelWithCorners ℝ F H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H' N] [IsManifold J (↑⊤) N] [T2Space N] [J.Boundaryless] [T2Space M] [I.Boundaryless] (ψ : Diffeomorph (I.prod (modelWithCornersSelf ℝ (E →L[ℝ] ℝ))) (J.prod (modelWithCornersSelf ℝ (F →L[ℝ] ℝ))) (TauCeti.CotangentBundle I M) (TauCeti.CotangentBundle J N) ↑⊤) : (∃ (h : Diffeomorph I J M N ↑⊤), h.cotangent = ψ) ↔ (TauCeti.SmoothForm.mpullback ⇑ψ ⋯) (TauCeti.CotangentBundle.liouvilleForm J N) = TauCeti.CotangentBundle.liouvilleForm I M

Characterization of cotangent lifts. On Hausdorff manifolds without boundary, a smooth diffeomorphism ψ : T*M → T*N is the cotangent lift of a smooth diffeomorphism M → N exactly when it pulls the Liouville form λ_N back to λ_M.