Reeb Vector Field in Contact Geometry

• Reeb vector field is a unique, nowhere-vanishing vector field determined by a contact form that satisfies α(R) = 1 and ι_R dα = 0 on odd-dimensional manifolds.
• It underpins contact topology by ensuring geodesibility, invariant volume properties, and structuring global dynamics such as periodic Reeb orbits and Legendrian chords.
• The Reeb–Beltrami correspondence links contact geometry with fluid dynamics, enabling applications that range from Beltrami fields to geodesic flow analysis on curved manifolds.

A Reeb vector field is a fundamental object in contact geometry, providing the canonical direction transverse to a contact distribution on an odd-dimensional manifold. Precisely, for any contact form $\alpha$ on a $(2n+1)$-dimensional smooth manifold $M$, the Reeb vector field $R_\alpha$ is uniquely determined by $\alpha(R_\alpha)=1$ and $d\alpha(R_\alpha,\cdot)=0$. This construction is central not only to the theory of contact manifolds but also extends to diverse domains such as fluid dynamics via Beltrami fields, global dynamical systems, and the topology of odd-dimensional manifolds.

1. Characterization and Fundamental Properties

Given a contact form $\alpha\in\Omega^1(M)$ on an oriented smooth manifold $M^{2n+1}$, $\alpha$ must satisfy $\alpha\wedge(d\alpha)^n>0$ everywhere. The Reeb vector field $R_\alpha$ is then the unique nowhere-vanishing vector field fulfilling: $$\alpha(R_\alpha) = 1,\qquad \iota_{R_\alpha} d\alpha = 0,$$ i.e.\ $R_\alpha$ generates the characteristic foliation transverse to the contact hyperplane field $\xi=\ker\alpha$ (Hajduk et al., 2011, Geiges, 2020, Colin et al., 20 Jan 2025, Becker, 2024).

Every Reeb vector field is geodesible: there exists a Riemannian metric for which the flow lines of $R_\alpha$ are geodesics, parametrized by arc-length. Sullivan's classical equivalence states that $R_\alpha$ is geodesible if and only if it admits a connection form $\eta$ with $\eta(R_\alpha)=1$ and $\iota_{R_\alpha}d\eta=0$ (Hajduk et al., 2011, Geiges, 2020, Becker, 2024).

2. Reeb Fields and Contact Topology

Basic Cohomological Invariants

Contact Reeb fields are distinguished by the nonvanishing of their basic cohomology class $[d\alpha]\in H^2_{\mathrm{bas}}(M,\xi)$, where forms $\omega$ satisfy $\iota_{R_\alpha}\omega=0$, $\iota_{R_\alpha}d\omega=0$. The Euler class $e(R_\alpha):=-[d\alpha]$ in basic cohomology is topologically significant: $$\text{If } e(R_\alpha)=0, \text{ there exists a foliation transverse to } R_\alpha; \text{ further, } M \text{ fibers over } S^1.$$ Conversely, if $e(R_\alpha)\neq0$, the contact structure is not transversely integrable (Hajduk et al., 2011, Geiges, 2020).

Existence on Odd-Dimensional Manifolds

The Hajduk–Walczak construction demonstrates that on every closed, oriented $(2n+1)$-manifold, one can build a vector field that is geodesible and carries a nontrivial basic class---precisely the two hallmark properties of contact Reeb fields. Open-book decompositions are employed to glue local models and connection forms from lower-dimensional bindings, ultimately ensuring the existence of a contact structure on every such manifold (Hajduk et al., 2011).

Volume and Diffeomorphism

Distinct contact forms sharing the same Reeb vector field induce the same volume: $$\int_M \alpha_0\wedge(d\alpha_0)^n = \int_M \alpha_1\wedge(d\alpha_1)^n,$$ if both $\alpha_0,\alpha_1$ have $R$ as their Reeb vector field (Geiges, 2020). The volume is topologically invariant under isotopies of the Reeb flow.

3. Dynamical Systems and Reeb Orbits

The periodic orbits and chords of the Reeb flow form the core of contact dynamics. In three dimensions, the periodic Reeb trajectories ("Reeb orbits") encode much of the global topology and contact invariants.

Global Surfaces of Section

For generic contact forms on a closed $3$-manifold, every hyperbolic periodic Reeb orbit admits transverse homoclinic connections. There always exists a global surface of section (Birkhoff section) with prescribed boundary periodic Reeb orbits and prescribed interior Legendrian links, enabling a reduction of the dynamics to area-preserving return maps on surfaces (Colin et al., 20 Jan 2025).

Reeb Chords of Legendrian Knots

Any Legendrian knot admits infinitely many Reeb chords for generic contact forms, except in a small set of exceptional cases (lens spaces, $S^3$ with only two Reeb orbits). This imposes severe constraints on the knotting and linking structure realized by Reeb dynamics (Colin et al., 20 Jan 2025).

4. The Reeb–Beltrami Correspondence and Fluid Flows

A key analytic correspondence exists between Reeb vector fields and Beltrami fields in fluid dynamics. On a $3$-manifold $(M,g)$, a nowhere-vanishing, divergence-free Beltrami field $X$ (i.e.\ $\mathrm{curl}\, X = \lambda X$ for some $\lambda\neq0$) induces a contact form $\alpha=i_X g$; $X$ is, up to rescaling, the Reeb field of $\alpha$ (Fontana-McNally et al., 2023). Conversely, every Reeb-like pair $(X,\alpha)$ is (for a suitable metric) a Beltrami pair.

This correspondence is equivariant under symmetries: given a group action preserving the Beltrami pair, the associated contact form and Reeb field inherit the symmetry (Fontana-McNally et al., 2023).

The Kepler–Euler flow, i.e.\ the lifted geodesic flow on the spherical cotangent bundle of a constant curvature surface, realizes the regularized Kepler problem as a Reeb (and simultaneously Beltrami) field on the associated manifold.

5. Geodesic Reeb Fields, Space Forms, and Tightness

On a Riemannian $3$-manifold, a unit geodesic vector field $X$ is the Reeb field of its metric dual $1$-form $\alpha=i_X g$ if $\alpha$ is a contact form, i.e.\ if the shape operator $\beta: X^\perp \to X^\perp$, $v \mapsto \nabla_v X$, is nowhere self-adjoint. This is equivalent to the absence of Jacobi fields vanishing at two points unless curvature imposes further restrictions (Becker, 2024, Harris et al., 2013).

On constant-curvature manifolds:

• All unit geodesic fields are Reeb in positive curvature ($S^3$);
• In flat case, Reeb coincides with geodesic only for non-parallel fields with $\beta\neq0$ everywhere (on tori, tight contact structures are indexed by volume);
• The induced contact structure is universally tight if the Reeb flow is periodic, isometric, or free and proper (Becker, 2024).

In Sasakian manifolds, the Reeb field is a unit Killing field satisfying $\nabla_X\xi=-\phi X$ for the contact metric structure $(\phi,\xi,\eta,g)$, and controls the ambient geometry and explicit curvature identities (Degla et al., 2010).

6. Reeb Flows, Right-Handedness, and Contact-Type Extensions

On closed $3$-manifolds, any volume-preserving right-handed vector field (all orbits link positively) is conformally Reeb: its associated closed 2-form $\omega=\iota_X\Omega$ is contact-type, i.e.\ admits a contact form $\alpha$ (with $d\alpha=\omega$) such that after rescaling, $X$ becomes the Reeb field (Prasad, 2022). This is established via linking integrals and applies powerful McDuff–Ghys criteria. Such flows admit global surfaces of section and always have at least two simple periodic orbits by the Taubes–Weinstein theorem.

7. Special Geometric Applications and Examples

• Real hypersurfaces of type A in complex Grassmannians are classified via invariance conditions on the Reeb vector field and associated distributions under the shape operator (Lee et al., 2015).
• In great circle flows on $S^3$, every such flow is the Reeb flow of its metric dual form; volume-preserving flows must be Hopf (Harris et al., 2013, Becker, 2024).
• In almost contact B-metric manifolds, the Reeb vector field may be a vertical torse-forming field and play the role of Yamabe soliton potential under contact-conformal transformations, yielding explicit structures in both main and cosymplectic classes (Manev, 2022).

Table: Characteristic Properties of Reeb Vector Fields

Property	Criterion/Formula	Reference
Contact-criterion	$\alpha \wedge (d\alpha)^n > 0$, Reeb: $\alpha(R)=1$, $\iota_R d\alpha=0$	(Hajduk et al., 2011, Geiges, 2020)
Geodesibility	Existence of connection form, or suitable metric	(Hajduk et al., 2011, Becker, 2024)
Basic cohomology class	$[d\alpha]\neq0$ in $H^2_{\mathrm{bas}}(M,\xi)$	(Hajduk et al., 2011)
Equivalence to Beltrami field	$\mathrm{curl}_g X = \lambda X$, $X$ divergence-free	(Fontana-McNally et al., 2023)
Volume invariance	$\int_M \alpha\wedge(d\alpha)^n$	(Geiges, 2020)
Universal tightness	Periodic/isometric/free-proper Reeb flows	(Becker, 2024)
Right-handedness	Conformally Reeb after rescaling	(Prasad, 2022)

The Reeb vector field unifies the analytic, topological, and dynamical aspects of contact geometry, serving as the central vehicle for exploring symplectic fillings, surface dynamics, and flows with deep topological constraints. Its basic invariants, correspondence to fluid flows, and role in global dynamical decomposition make it indispensable in modern geometric analysis and topology.