Bi-Contact Structures in 3-Manifolds

• Bi-contact structures are paired, transverse positive and negative contact structures whose intersection forms a line field supporting projectively Anosov flows.
• They offer a framework for classification using Liouville pairs, differential invariants, and local normal forms derived via the Cartan equivalence method.
• Applications include hyperbolic flow surgery, geodesic flow constructions on hyperbolic surfaces, and explicit models on T³ linking contact topology with symplectic fillings.

A bi-contact structure on a 3-manifold is a pair of transverse, cooriented contact structures—one positive, one negative—intersecting in a distinguished line field whose dynamics interface rich areas of contact geometry, symplectic topology, and hyperbolic dynamics. Bi-contact geometry gives foundational characterizations of (projectively) Anosov flows, provides invariants and normal forms via the Cartan equivalence method, supports a surgery theory for modifying flows, and acts as a framework linking contact topology with the ergodic theory of flows. Recent developments have yielded classification results, operational criteria, and a robust suite of examples relating Reeb flows, Liouville structures, and contact/symplectic fillings.

1. Definitions and Foundational Constructions

A contact structure on an oriented 3-manifold $M$ is a maximally non-integrable cooriented 2-plane field, expressible locally as $\xi = \ker \alpha$ for a 1-form $\alpha$ with $\alpha \wedge d\alpha \neq 0$ everywhere. If $\alpha \wedge d\alpha > 0$ (resp. < 0), $\xi$ is called positive (resp. negative).

A bi-contact structure is a pair $(\xi_-, \xi_+)$ of contact structures, with $\xi_+ = \ker \alpha_+$ positive and $\xi_- = \ker \alpha_-$ negative, such that $\xi_- \oplus \xi_+ = TM$; the distributions are everywhere transverse and their intersection defines a line field $\xi = \ker \alpha$0 (Hozoori, 2020Klotz et al., 2023Salmoiraghi, 2021Barthelmé, 11 Feb 2025).

The defining 1-forms for a bi-contact structure must satisfy: $\xi = \ker \alpha$1 at every point. Bi-contact structures exhibit local Darboux forms: around any point, coordinates $\xi = \ker \alpha$2 can be chosen with

$\xi = \ker \alpha$3

or, equivalently, $\xi = \ker \alpha$4, $\xi = \ker \alpha$5.

A bicontactomorphism is a diffeomorphism preserving both distributions, i.e., for $\xi = \ker \alpha$6, there exist nowhere-zero functions $\xi = \ker \alpha$7 with $\xi = \ker \alpha$8, $\xi = \ker \alpha$9 (Klotz et al., 2023).

2. Characterizations via Anosov and Projectively Anosov Flows

A (projectively) Anosov vector field $\alpha$0 on $\alpha$1 is one whose linear flow on the normal bundle admits a continuous dominated splitting with distinguished expansion and contraction rates (Hozoori, 2020Barthelmé, 11 Feb 2025Hozoori, 2021). Every projectively Anosov flow naturally supports a bi-contact structure: $\alpha$2 is tangent to the intersection $\alpha$3. Each of the associated contact planes is a small perturbation of the co-unstable and co-stable 2-planes, with positivity/negativity aligned to the classical splitting.

Hozoori's theorem provides: A flow $\alpha$4 (with generator $\alpha$5) on $\alpha$6 is Anosov if and only if there exists a bi-contact structure $\alpha$7 with contact 1-forms $\alpha$8 such that both $\alpha$9 and $\alpha \wedge d\alpha \neq 0$0 are Liouville pairs; that is, the 2-forms

$\alpha \wedge d\alpha \neq 0$1

and its twisted analog are symplectic on $\alpha \wedge d\alpha \neq 0$2 (Hozoori, 2020). This Liouville criterion recovers all classical Anosov expansion/contraction estimates. The summary table below organizes the relationships:

Structure	Condition	Role
Bi-contact Structure	$\alpha \wedge d\alpha \neq 0$3	Supports projectively Anosov flows
Liouville Pair	$\alpha \wedge d\alpha \neq 0$4	Ensures Anosov property
Anosov Flow	Hyperbolic splitting	Implies (and implied by) bi-contact

3. Reeb Dynamics, Differential Invariants, and Liouville Pairs

For any contact form $\alpha \wedge d\alpha \neq 0$5, the Reeb vector field $\alpha \wedge d\alpha \neq 0$6 satisfies $\alpha \wedge d\alpha \neq 0$7 and $\alpha \wedge d\alpha \neq 0$8. The dynamical characterization of Anosovity is recast: A projectively Anosov $\alpha \wedge d\alpha \neq 0$9 is Anosov iff some supporting bi-contact structure admits a Reeb field (for either contact form) that is "bitransverse," i.e., transverse to both invariant subbundles $\alpha \wedge d\alpha > 0$0, $\alpha \wedge d\alpha > 0$1 (Hozoori, 2020Barthelmé, 11 Feb 2025). If $\alpha \wedge d\alpha > 0$2 is dynamically negative and coincides, up to reparametrization, with $\alpha \wedge d\alpha > 0$3, the Anosov property holds.

A Liouville pair $\alpha \wedge d\alpha > 0$4 on $\alpha \wedge d\alpha > 0$5 ensures that the associated 2-form $\alpha \wedge d\alpha > 0$6 extends to a symplectic filling of the disjoint union $\alpha \wedge d\alpha > 0$7. This symplectic cobordism encodes exactness and fillability properties vital to symplectic and contact topology (Hozoori, 2020Hozoori, 2021).

Differential invariants: Using Cartan's method, bi-contact structures admit a torsion function $\alpha \wedge d\alpha > 0$8, and associated invariants $\alpha \wedge d\alpha > 0$9, $\xi$0, and a directional derivative $\xi$1, which classify local normal forms up to bicontactomorphism (Klotz et al., 2023). The quadratic form

$\xi$2

distinguishes between contact ellipses ($\xi$3), contact hyperbolas ($\xi$4 or $\xi$5), and degenerate cases.

4. Local and Global Normal Forms, Symmetries, and Invariant Structures

Local normal forms for bi-contact pairs with symmetry are classified: When a nonzero infinitesimal symmetry exists, seven normal forms arise, varying by the invariants $\xi$6 (generating invariant) and the Schwarzian $\xi$7 (Jackman, 1 Aug 2025). Around any point of an orientable Anosov flow, local models are linear: $\xi$8, $\xi$9, with symmetry algebra computed accordingly.

For generic bi-contact pairs, symmetries are typically minimal (one-dimensional), reflecting rigidity: maximal finite-dimensional or infinite-dimensional symmetry algebras occur only in specific, highly structured "flat" models. Along the axis field $(\xi_-, \xi_+)$0, the planes restrict to Legendrian foliations carrying a canonical projective structure, compared via the Schwarzian invariant (Jackman, 1 Aug 2025).

By the Cartan equivalence method, normal forms and adapted Cartan e-structures are established (Klotz et al., 2023). In particular, the structure equations allow identification of "symplectic prolongations," natural Riemannian metrics, Levi-Civita connections, and computations of sectional/scalar curvatures.

5. Bi-Contact Surgery and Modifications of Hyperbolic Flows

Salmoiraghi's and Goodman’s surgery constructions use bi-contact geometry to modify flows in neighborhoods of periodic orbits. Bi-contact surgery involves locally replacing a bi-contact pair $(\xi_-, \xi_+)$1 in a flow-box with new forms $(\xi_-, \xi_+)$2 of prescribed shape, maintaining positivity/negativity, transversality, and compatibility with global structure (Hozoori, 2021Salmoiraghi, 2021). The surgery produces a new bi-contact structure supporting a (projectively) Anosov flow on the surgered manifold.

For any Anosov flow and any periodic orbit, such surgery can be performed in an arbitrarily small neighborhood, fully recovering the Goodman surgery as a special case (Hozoori, 2021). The construction is tightly constrained by contact positivity conditions (involving derivatives of the local defining functions) and matching conditions along the surgery region’s boundary.

Applications to geodesic flows on unit tangent bundles of hyperbolic surfaces show that contact Anosov flows can be generated by Foulon–Hasselblatt Legendrian surgery if and only if the surgery is performed along a simple closed geodesic (Salmoiraghi, 2021).

6. Uniqueness, Isotopy, and Classification Results

Supporting bi-contact structures for a given projectively Anosov flow are unique up to homotopy through supporting structures (Hozoori, 2020). If the manifold $(\xi_-, \xi_+)$3 is atoroidal (e.g., hyperbolic), any two supporting positive contact structures are isotopic through supportings. The only obstruction to such isotopy in torus suspensions is the presence of Giroux torsion, detected by criteria from strong symplectic fillability.

Classifications distinguish three scenarios for Anosov-supporting bi-contact pairs (Barthelmé, 11 Feb 2025):

1. The positive structure admits a bitransverse Anosov Reeb, yielding a positively skew Anosov flow.
2. The negative structure admits such a Reeb, yielding a negatively skew flow.
3. Neither is Anosov-contact, so the flow is either a suspension or not $(\xi_-, \xi_+)$4-covered.

If the Anosov flow is skew $(\xi_-, \xi_+)$5-covered, every supporting contact structure is, up to isotopy, the canonical one (Hozoori, 2020). The dynamical implications and periodic orbit properties of distinct Reeb flows are accessible via the periodic homotopy data and cylindrical contact homology.

7. Examples and Applications

• Geodesic flows: The unit tangent bundle of a closed negatively curved surface admits canonical contact forms giving rise to a bi-contact structure; the geodesic flow is the Reeb flow of one (Hozoori, 2020).
• Torus suspensions: Suspensions of hyperbolic toral automorphisms yield explicit contact 1-forms forming a Liouville pair after appropriate adjustment.
• Examples on $(\xi_-, \xi_+)$6: Contact structures of the form $(\xi_-, \xi_+)$7 for $(\xi_-, \xi_+)$8 and small $(\xi_-, \xi_+)$9 yield tight, transverse structures whose intersection is projectively Anosov, with Liouville pair construction providing the only obstruction to full Anosovity (Hozoori, 2020).

In each case, the local and global geometry, dynamics, and symplectic fillability are deeply informed by the intrinsic structure of the bi-contact pair, the arrangement of its Reeb dynamics, and the associated invariants.

Summary:

Bi-contact structures provide a fundamental unifying mechanism for the study of Anosov and projectively Anosov flows in dimension three. Their geometric and dynamical properties are intertwined with contact topology, symplectic fillings, and rigid local and global classification results. Recent advances have systematized their invariants, clarified their role in surgery constructions, and outlined conjectural frameworks for their classification, highlighting the centrality of bi-contact topology in the geometry of 3-manifolds (Hozoori, 2020Klotz et al., 2023Jackman, 1 Aug 2025Hozoori, 2021Salmoiraghi, 2021Barthelmé, 11 Feb 2025).