Reeb fields of energy hypersurfaces

Let σ be a nondegenerate two-form on a finite-dimensional manifold M, let θ be a one-form with dθ = -σ, and let ι : S → M be a hypersurface on which a function K is constant, in the sense that the tangent image of ι at s is exactly the kernel of dK. If the restriction ι^* θ is a contact form, then its Reeb vector field is a rescaling of the Hamiltonian vector field X_K of K:

dι (R_(ι^* θ)) = θ(X_K)⁻¹ • X_K

(TauCeti.SmoothTwoForm.IsNondegenerate.mfderiv_reebVectorField_mpullback), wherever θ(X_K) does not vanish. Indeed X_K is tangent to the level, since dK(X_K) = σ(X_K, X_K) = 0, and for every vector w tangent to the level, dθ(X_K, w) = -σ(X_K, w) = -dK(w) = 0. So the lift of X_K / θ(X_K) satisfies the two Reeb equations, which determine the Reeb vector field.

This is how Reeb flows on hypersurfaces of contact type are reparametrized Hamiltonian flows: on the unit cosphere bundle of a Riemannian manifold, with θ the Liouville form and K the kinetic energy, the factor θ(X_K) is identically 1, and the Reeb flow is the cogeodesic flow.

References

• D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Section 3.5 (hypersurfaces of contact type and their characteristic line fields).
• H. Geiges, An Introduction to Contact Topology, Cambridge University Press, 2008, Section 1.4 (Reeb vector fields).

theorem TauCeti.SmoothTwoForm.IsNondegenerate.mfderiv_reebVectorField_mpullback {E : Type u_1} {H : Type u_2} {M : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] {E' : Type u_4} {H' : Type u_5} {S : Type u_6} [NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] [TopologicalSpace H'] {J : ModelWithCorners ℝ E' H'} [TopologicalSpace S] [ChartedSpace H' S] [IsManifold J 2 S] {σ : SmoothTwoForm I M} (hσ : σ.IsNondegenerate) {θ : SmoothForm I M ℝ 1} {ι : S → M} (hι : ContMDiff J I (↑⊤) ι) (hα : ((SmoothForm.mpullback ι hι) θ).IsContactForm) {K : M → ℝ} {s : S} (hdθ : mextDeriv (↑θ) (ι s) = -↑σ (ι s)) (hK : (↑(mfderiv% ι s)).range = (↑(d% K (ι s))).ker) (hθK : (↑θ (ι s)) ![σ.hamiltonianVectorField K (ι s)] ≠ 0) : (mfderiv% ι s) (((SmoothForm.mpullback ι hι) θ).reebVectorField s) = ((↑θ (ι s)) ![σ.hamiltonianVectorField K (ι s)])⁻¹ • σ.hamiltonianVectorField K (ι s)

The Reeb field of an energy hypersurface is a rescaled Hamiltonian field. Let σ be a nondegenerate two-form and θ a one-form with dθ = -σ at ι s, and suppose that the tangent image of ι : S → M at s is the kernel of dK. If the restriction ι^* θ is a contact form and θ(X_K) ≠ 0 at ι s, then the differential of ι sends the Reeb vector field of ι^* θ to X_K / θ(X_K).