The unit cosphere bundle is of restricted contact type

Let M be a smooth Riemannian manifold without boundary, with cotangent bundle T*M = TauCeti.CotangentBundle I M, Liouville form λ = p dq and canonical symplectic form ω = -dλ. The radial field Z = p ∂_p, the Euler vector field of the bundle, is the Liouville field of the primitive -λ of ω (TauCeti.CotangentBundle.symplecticForm_interior_vectorBundleEulerField).

Along the fibre ray t ↦ t • z the norm of covectors is t ↦ |t| ‖z.2‖, so the differential of the norm sends Z to ‖z.2‖ (TauCeti.CotangentBundle.mfderiv_norm_snd_vectorBundleEulerField). On the unit cosphere bundle S*M = {‖p‖ = 1}, a regular level of the norm, this is 1 ≠ 0, so Z is transverse to S*M (TauCeti.CotangentBundle.isTransverseAlong_vectorBundleEulerField_cosphereBundle). Hence S*M is a hypersurface of restricted contact type in (T*M, ω), with primitive -λ (TauCeti.CotangentBundle.isRestrictedContactType_cosphereBundle). This is the model case of the hypersurfaces of contact type in cotangent bundles whose closed characteristics are produced from almost existence.

Consequently the restriction α_b = -λ|S*M of the primitive to S*M is a contact form (TauCeti.CotangentBundle.isContactForm_mpullback_neg_liouvilleForm_cosphereBundle), and so is its negative α_g = λ|S*M, which defines the opposite coorientation of the same contact distribution (TauCeti.CotangentBundle.isContactForm_mpullback_liouvilleForm_cosphereBundle).

Main results

• TauCeti.CotangentBundle.isTransverseAlong_vectorBundleEulerField_cosphereBundle: p ∂_p is transverse to S*M.
• TauCeti.CotangentBundle.isRestrictedContactType_cosphereBundle: S*M is of restricted contact type, with primitive -λ and Liouville field p ∂_p.
• TauCeti.CotangentBundle.isContactForm_mpullback_neg_liouvilleForm_cosphereBundle, TauCeti.CotangentBundle.isContactForm_mpullback_liouvilleForm_cosphereBundle: -λ|S*M and λ|S*M are contact forms.

References

• D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Section 3.5 (the unit cotangent bundle as a hypersurface of contact type).
• J.-F. Barraud, M. Damian, V. Humilière, A. Oancea, Floer homology with DG coefficients. Applications to cotangent bundles, arXiv:2404.07953, §1.2.

theorem TauCeti.CotangentBundle.isTransverseAlong_vectorBundleEulerField_cosphereBundle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] : IsTransverseAlong (modelWithCornersSelf ℝ (Fin (Module.finrank ℝ (E × (E →L[ℝ] ℝ)) - 1) → ℝ)) (vectorBundleEulerField I) Subtype.val

The radial field p ∂_p is transverse to the unit cosphere bundle S*M: the differential of the norm of covectors sends it to 1 along S*M.

theorem TauCeti.CotangentBundle.isRestrictedContactType_cosphereBundle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] : IsRestrictedContactType (modelWithCornersSelf ℝ (Fin (Module.finrank ℝ (E × (E →L[ℝ] ℝ)) - 1) → ℝ)) (↑(symplecticForm I M)) (-↑(liouvilleForm I M)) Subtype.val

The unit cosphere bundle is of restricted contact type. For the canonical symplectic form ω = -dλ of the cotangent bundle of a Riemannian manifold, S*M is a hypersurface of restricted contact type with primitive -λ: its Liouville field is the radial field p ∂_p, which is transverse to S*M.

theorem TauCeti.CotangentBundle.isContactForm_mpullback_neg_liouvilleForm_cosphereBundle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] : ((SmoothForm.mpullback Subtype.val ⋯) (-liouvilleForm I M)).IsContactForm

The unit cosphere bundle is a contact manifold. The restriction α_b = -λ|S*M of the primitive -λ of the canonical symplectic form ω = -dλ to the unit cosphere bundle S*M is a contact form.

theorem TauCeti.CotangentBundle.isContactForm_mpullback_liouvilleForm_cosphereBundle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] : ((SmoothForm.mpullback Subtype.val ⋯) (liouvilleForm I M)).IsContactForm

The restriction α_g = λ|S*M of the Liouville form to the unit cosphere bundle, which defines the opposite coorientation of the contact distribution of α_b = -λ|S*M, is a contact form.