Contact geometry and Reeb flows

On a smooth (2n+1)-manifold, a global one-form α is contact when α∧(dα)^n never vanishes. Its kernel ξ is a cooriented contact distribution of dimension 2n, and dα restricted to ξ is nondegenerate. For a general hyperplane field, integrability would require the Frobenius condition; in dimension three this includes α∧dα=0. The contact condition takes the opposite extreme. It does not say that dα is nondegenerate on the full odd-dimensional tangent space, which would be impossible.

The Reeb field R_α is characterized by α(R_α)=1 and ι_{R_α}dα=0. The kernel of dα is one-dimensional, and contactness makes α nonzero on it, giving existence and uniqueness after normalization. Cartan’s identity then gives L_{R_α}α=0, so the Reeb flow preserves both α and its contact volume wherever the flow exists. On a noncompact manifold it need not be complete. Its trajectories are geometric data associated with the chosen form, rather than an arbitrary vector field tangent to the contact hyperplanes.

Multiplying α by a positive function leaves its kernel unchanged and preserves the contact condition. It need not merely rescale the original Reeb field: differentiating the function adds a term df∧α to d(fα), so the kernel of that derivative can tilt. A positive constant rescales R by its reciprocal, but a nonconstant factor can add a component in the contact hyperplanes. Therefore a contact structure and a Reeb dynamical system are not interchangeable specifications. Preserving the distribution need not preserve one chosen Reeb flow.

For any contact form, the product M×R with coordinate s has symplectic form d(exp(s)α). Expanding its top wedge gives a nonzero multiple of ds∧α∧(dα)^n, proving nondegeneracy; closedness follows by construction. This symplectization is exact and noncompact in the extra direction. It converts the contact distribution into an even-dimensional geometry, but it does not identify all possible Reeb choices as the same vector field. The positive scaling of α corresponds to changing the hypersurface or section used inside that symplectization.

A hypersurface Σ in a symplectic manifold is of contact type when a vector field Y defined near it satisfies L_Yω=ω and is transverse to Σ. Then α=(ι_Yω)|_Σ is a contact form and dα=ω|_Σ. Transversality ensures the relevant top wedge is nonzero; simply being a regular energy level does not by itself supply such a Y. For Σ={H=E} regular, the Hamiltonian field lies in the kernel of the restricted two-form. On a contact-type level, R_α=X_H/α(X_H), with the sign and speed fixed by that denominator.

Under our convention ι_{X_H}ω=dH, the denominator equals −dH(Y). This sign is easy to reverse if one imports a formula from a text with the opposite Hamiltonian convention. Closed Reeb orbits are central to contact topology, but their existence requires theorem hypotheses; no assertion that every Reeb trajectory is periodic or that every contact manifold has the same orbit behavior is made here. Computing the field and checking the contact condition comes before using any global dynamical conclusion.