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Goodman surgery and projectively Anosov flows

We introduce a generalization of Goodman surgery to the category of projectively Anosov flows. This construction is performed along a knot that is simultaneously Legendrian and transverse for a supporting bi-contact structure. When the flow is Anosov, our operation generates the same flows of Goodman's construction. The Anosovity of the new flow is strictly connected to contact geometry. We use this relation to give a refinement to the sequence of surgeries coefficients producing Anosov flows in concrete examples. We give an interpretation of the bi-contact surgery in terms of contact-Legendrian surgery and admissible-inadmissible transverse surgery and we deduce some (hyper)tightness result for contact and transverse surgeries. Outside of the realm of Anosov flows, we produce new projectively Anosov flows on hyperbolic 3-manifolds with leaves of genus g>1 in the invariant foliations.


Introduction
A classic example of the beautiful intertwine between hyperbolic dynamics, foliation theory, and contact geometry is given by an Anosov flow. Geometrically an Anosov flow is determined by two transverse invariant foliations with expanding and contracting behaviors [Ano63], [Ano67]. In the mid 90s Mitsumatsu [Mit95] first noticed that every Anosov vector field with orientable weak invariant foliations also belongs to the intersection of a pair of transverse contact structures rotating towards each other. We call these pairs bi-contact structures and the associated flows projectively Anosov flows (see also [ET98]). Contrarily to Anosov flows, projectively Anosov flows are abundant: there is a multitude of 3-manifolds that carry bi-contact structures but do not carry any Anosov flow.
The importance of projectively Anosov flows makes the introduction of surgery operations desirable. In this paper we define a surgery operation on a annulus transverse to a projectively Anosov flow. To this end, we identify a projectively Anosov flow with a supporting bi-contact structure. From now on we assume q integer.
Theorem 1. Given a Legendrian-transverse knot K in a bi-contact structure on a 3manifold, there is a (1, q)-Dehn type surgery on a transverse annulus containing K, such that for infinitely many values of q it yields bi-contact structures (or equivalently projectively Anosov flows) in the new manifold.
The bi-contact surgery of Theorem 1 can be interpreted as a natural extension to projectively Anosov flows of the surgery operation on Anosov flows introduced by Goodman [Goo83]. Indeed, a recent result of Hozoori [Hoz21a] shows that the operation of Theorem 1 can be performed under the same hypothesis of Goodman's construction and it generates the same flows.
The Anosovity of the flows constructed by Goodman surgery relies on the existence of a cone field satisfying a number of conditions according to the cone criterion for hyperbolicity. These conditions translate to bi-contact geometry and allow us to construct Lorentzian metrics whose positive cones contract 'on the nose'. We use these metrics to show the following.

Theorem 2. Let M be a 3-manifold endowed with an Anosov flow with orientable weak invariant foliations. The surgery of Theorem 1 yields an Anosov flow if and only if it yields a projectively Anosov flow.
The original construction of Goodman is performed near a closed orbit and requires the choice of an annulus that is transverse to the flow. This comes with a preferred direction, meaning that surgeries in that direction yield Anosov flows. Since in a neighborhood of a closed orbit there are always annuli with opposite preferred directions, Goodman surgery yields Anosov flows for every q ∈ Z. It was only few years after the pubblication of Goodman's work that the experts realized that her construction can be performed along a knot transverse to the weak stable and unstable manifolds [Bar06], and the existence of a closed orbit in a neighborhood is not necessary. However, under these hypothesis, the condition on the direction of the surgery that produces Anosov flows can not be bypassed and there are surgery coefficients that do not yield hyperbolic flows. The bi-contact formulation of the cone criterion allows us to obtain additional information about the sequence of the surgeries that yields Anosov flows. We give an explicit computation using the following.
The surgery coefficient 1/q is defined with respect to the framing induced by ξ. Note that since an inadmissible transverse surgery adds twisting to the original contact structure, there is no reason to expect that such operation yields, in general, a tight contact structure.
A theorem of Conway [Con19] allows us to translate the above result in terms contact surgeries on a Legendrian push-off of K. In a neighborhood of a closed orbit we deduce the following.
Corollary 5. Let γ be a closed orbit in an Anosov flow with orientable weak invariant foliations. There is a positive supporting contact structure η such that every positive 1/q-contact surgery along γ yields a hypertight contact structureη.
A consequence of Corollary 5 is that every positive 1/q-contact surgery on any suspension knot of a T 2 -bundle with Anosov monodromy equipped with a vertically rotating positive contact structure with minimal twisting is hypertight.
The bi-contact structures defining Anosov flows have been fully characterized from a symplectic and contact-geometric prospective by Hozoori [Hoz20b].
In [BBP] Bonatti, Bowden and Potrie sketch the construction of a projectively Anosov flow in a hyperbolic 3-manifold that has attracting or repelling orbits (therefore not Anosov) with taut invariant foliations using an adaptation of the hyperbolic plugs as defined by Béguin, Bonatti and Yu in [BBY17].
Using Theorem 1 we construct new families of non-Anosov bi-contact structures on hyperbolic manifolds. Corollary 6. There are infinitely many hyperbolic manifolds that carry bi-contact structures with invariant foliations with leaves of genus g > 1.
The construction of Corollary 6 is roughly as follows. Mitsumatsu ([Mit95], [Mit02]) introduces a special family of tight bi-contact structures called propellers on every T 2bundles over the circle. For instance, consider the Anosov automorphism of T 2 induced by the linear map A = 2 1 1 1 and consider the torus bundle over the circle M (0) = T 2 × [0, 1]/ ∼, where the equivalence relation ∼ is defined by (x, 0) ∼ (A(x), 1). The complement of the knot K given by the suspension of (0, 0) is diffeomorphic to the complement of the figure-eight knot in S 3 . Call M (q) the manifold obtained by performing Dehn (1, q)-surgery on K in M (0). As a consequence of a celebrated result of Thurston [Thu83], M (q) is hyperbolic except when q ∈ {0, ±1, ±2, ±3, ±4}. It is easy to see that if M (0) is equipped with a propeller, K is always a Legendrian-transverse knot. Therefore, we can apply the surgery described in Theorem 1 to produce an infinite family of hyperbolic manifolds andowed with a bi-contact structure. We show that there exists a special choice of the propeller such that a bi-contact (1, q)-surgery yields exactly the manifolds M (q). Finally, the existence of leaves of genus g > 0 in the new invariant foliations is a consequence of the presence of exceptional compact leaves in the invariant foliations of the initial flow. This shows that the new flows are not Anosov.
1.1. Outline. In Section 2 we introduce the notion of bi-contact structure and projectively Anosov flow. In Section 3, Section 4, we review the definition of Dehn surgery in general 3-manifolds and 3-manifolds with structure (contact structures, Anosov flows). Section 5 is dedicated to the definition of the bi-contact surgery. In Section 6 we interpret the surgery in the realm of Anosov flows while In Section 7 we give a version of the cone criterion using bi-contact geometry. Section 8 is focused on the applications to contact geometry and finally in Section 9 we construct new non-Anosov bi-contact structures on hyperbolic manifolds.
1.2. Aknowledgment. The author would like to thank Tali Pinsky for her support and for introducing him to this fascinating subject. He would also like to thank Surena Hozoori and Martin Mion-Mouton for the helpful conversations. Finally the author would like to thank Ilaria Patania for her advice that greatly improved the quality of the manuscript.

Bi-contact structures and projectively Anosov flows
In the mid 90s Mitsumatsu [Mit95], and Eliashberg-Thurston [ET98], showed that an Anosov flow with orientable invariant weak foliation belongs to the intersection of two transverse contact structures. We call such a pair a bi-contact structure. However, the converse is not true and there are many bi-contact structures that do not define Anosov flows, as showed in the following example.
As Plante and Thurston show in [PT72], in a manifold that admit an Anosov the fundamental group grows exponentially and this of course not the case for T 3 .
Here · is induce by a Riemmanian metric on T M . We call E u and E s respectively the unstable bundle and the stable bundle.
The invariant bundles E u and E s induce invariant plane fields E u and E s on M . These plane fields are continuous and integrable, but unlike the Anosov case the integral manifolds my not be unique (see [ET98]). However, when they are smooth they also are uniquely integrable. We call these flows regular projectively Anosov (see [Nod04], [NT02] and [Asa04] for a complete classification). (Mitsumatsu [Mit95]). Let X be a C 1 vector field on M . Then X is projectively Anosov if and only if it is defined by a bi-contact structure.

Proposition 2.3
Example 2.4. What is mostly peculiar of Example 2.1 is the presence of compact leaves in the stable and unstable foliations (see Figure 1 on the right).
This construction can be extended to any T 2 -bundle over S 1 if we make the directions of the linear characteristic foliations of ξ = ker α and η = ker β on the top and bottom torus to agree with the rotation angles of the monodromy ( [Mit02] and see Figure 2). As Noda shows [Nod04], all the T 2 -bundles over S 1 with invariant foliations of class C 2 are obtained using this construction. Precisely, if the invariant foliations contains closed leaves, they are tori isotopic to linear flows on T 2 in the direction of the eigenvectors of the monodromy.
In the case the T 2 -bundle over S 1 has hyperbolic monodromy and the invariant foliations are of class C 2 without compact leaves, the flow is actually an Anosov flow given as the suspension of the monodromy. The bi-contact structures in these cases are obtained by propeller constructions with rotation angle less then π (see Figure 2).

A criterion to determine if a flow is Anosov.
We introduce Barbot's version [Bar06], [FH13] of the cone criterion for hyperbolicity.
3. Dehn type surgeries on contact 3-manifolds Surgery is a very effective way to produce new examples of manifold from classic ones. In this section we recall the basic notions and set the notation. We follow the descriptions and convention of [Gei08] and [BE13].
3.1. Dehn surgery along a knot. We call Dehn surgery along K the following construction. We first remove a tubolar neighborhood νK of a knot K, then we glue a solid torus Here p and q ∈ Z are coprime, µ is the meridian of ∂(νK) while λ is a preferred longitude (or framing of K) and we orient µ and λ so that µ ∩ λ = 1. We denote this operation as (p, q)-surgery or p/q-surgery.

Surgery along Legendrian knots.
If the ambient manifold is equipped with an additional geometric structure it is natural to ask for surgery operations that preserve it. For instance, given a contact 3-manifold (M, ξ) and a Legendrian knot K in M , it is possible to define a Dehn-type surgery along K that yield a new contact manifold (M ,ξ). We first choose the preferred longitude λ given by the contact frame. This can be done considering the push-off of K in a transverse direction to the contact planes along the knot. Every K has a neighborhood νK contactomorphic to a standard one as follows. Consider on R 2 × S 1 the tight contact structure ξ 1 = ker(dy − x dθ). Let N L the solid torus {(x, y, θ) : x 2 + y 2 ≤ 1}. This is the neighborhood of the Legendrian curve L = {(x, y, θ) : x = y = 0} and its boundary ∂N L is convex with two dividing curves parallel to Γ = {(x, y, θ) : x = 0, y = 1}. These solid tori are contactomorphic up to a perturbation of the boundary according to the following result Theorem 3.1 (Kanda [Kan97]). Let Γ be a pair of longitudinal curves on the boundary of a solid torus N . Let F be a (singular) foliation on ∂N that is divided by Γ. Then there is a unique (up to isotopy) tight contact structure on N whose characteristic foliation on ∂N is F.
We now remove a neighborhood νK of the Legendrian knot diffeomorphic to N L and glue back a solid torus S 1 × D 2 sending a meridian µ 0 = * × D 2 where * is a point of S 1 to a curve pµ + qλ on ∂(M \ νK). When we glue back the solid torus S 1 × D 2 we would like to extend the contact structure ξ on M \ νK over the solid torus. This boils down to finding contact structures on the solid torus S 1 × D 2 with a certain characteristic foliation on the convex boundary determined by the surgery coefficient p/q. When p = 0 this extension always exists and is tight in the solid torus by work of Honda and Giroux. If p = 1 (or equivalently for 1/q-surgery), we can assume that dividing set on the glued solid torus is a pair of curves parallel to the longitudinal curve qµ 0 + λ 0 where λ 0 = S 1 × * where * is a point of ∂D 2 . By Theorem 3.1 there is a unique tight contact structure on S 1 × D 2 and therefore a unique extension in the new manifold. We call this Dehn type surgery contact 1/q-surgery. Contact (−1)-surgery on K is historically called Legendrian surgery on K.
3.3. Surgery along transverse knots. A knot K in a contact manifold (M, ξ) is transverse if its oriented tangent vector is positively transverse to ξ. The natural framework to define transverse surgery has been set up by Gay in [Gay02], Conway in [Con19] , and Baldwin and Etnyre in [BE13]. This operation comes in two flavours, admissible transverse surgery and inadmissible transverse surgery both generalization of classic constructions introduced by Lutz and Martinet. Roughly speaking the first can be thought as removing twisting near a knot while the second as adding twisting.
A transverse knot has a neighborhood νK contactomorphic to a solid torus is an increasing surjective function of r (see fig []). If we identify some framing of K with λ = S 1 × {r = R, θ = 0}. The characteristic foliation on the torus r = R is given by parallel lines f (R)λ − (cot f (R))µ or equivalently lines of slope a = −(cot f (R))/f (R) where µ is a meridian of K. We adopt the convention that associates the slope −∞ to the contact planes transverse to the core of the torus (identified with K) while the 0 slope is associated with the slope of the preferred longitude λ. With this convention if we call N T a the solid torus with characteristic foliation of slope a ∈ (−∞, ∞) Note that different choices of the preferred longitude will give a different value of a.

Inadmissible transverse surgery.
Consider the open manifold obtained by removing K and than take the closure to obtain a manifold with boundary with a contact structure that induces on ∂M a characteristic foliation given by curves of slope −∞, or equivalently, meridional slope. We now glue a T 2 × I with rotational contact structure such that the contact planes rotate n half twists to some slope b, finally we perform a contact cut.

Goodman surgery on Anosov flows
We describe a Dehn-type surgery on Anosov flows introduced by Goodman [Goo83]. This construction, inspired by Handel-Thurston one [HT80], is very general and was used to construct the first family of Anosov flows on hyperbolic manifolds. We refer to [Sha] for a more detailed description. 4.1. Definition of Goodman surgery. Let X be a C 1 Anosov vector field on a 3manifold and consider a closed orbit γ of the flow with orientable invariant manifolds. A sufficiently small tubolar neighborhood of this orbit is cut by the invariant weak manifolds in four different sectors. In each of this sectors there is a smoothly embedded compct annulus C transverse to the flow and parallel to the invariant weak manifolds containing γ. The closed orbit γ can be enclosed in a neighborhood N C with smooth torus boundary such that C ⊂ ∂N C and the flow points outwards N C . We set M C = M \ int(N C ). Goodman first splits M in two pieces M C and N C and gluing back them together using a diffeomorphism φ : ∂N C → ∂M C with support in C. On the surgery annulus C there is a natural coordinate system given by two sets of curves described by parameters (l, m). The l-curves are closed and parallel to ∂C while the m-curves have endpoints on the boundary. The diffeomorphism can be described by a shear on C.

Anosovity of the new flows.
Anosovity is ensured for q > 0 if we choose an annulus in the first and third quadrant, and for q < 0 if the annulus is choosen in ther second or forth quadrant. The proof relies on the cone criterion for hyperbolicity (see Subsection 2.1 for a version that uses Lyapunov-Lorentz metrics) and consists to show the existence of a cone field that is positively invariant and exponentially contracted along the flow. The existence of annuli with different preferred directions in a neighborhood of a closed orbit, implies that Goodaman surgery can be performed for every q ∈ Z.
Remark 4.1. Let γ be a periodic orbit in an Anosov flow with orientable weak foliations. Fried [Fri83] introduced an operation that consists in blowing-up M along the normal bundle of γ obtaining a torus boundary component T γ with four periodic orbits, two attracting and two repelling. On T γ choose a foliation by circles C intersection the periodic orbits in exactly four points. then define the manifold M by collapsing these circles to points. The resulting flow is topologically Anosov because the foliations induced by the flow on M and the new manifoldM are the same outside the periodic orbit, and the new periodic orbitγ is clearly hyperbolic. However, it has been an open problem for more than thirty years to show that every flow flow obtained by Fried's surgery is orbit equivalent to differentiable Anosov flow (see [Bar17]). Recently Shannon [Sha] showed that this is actually the case in the category of transitive flows. A consequence of this fact is that the orbit equivalence classes of flows generated by Goodman depend on the surgery parameter, in the sense that a (1, q)-Goodman surgery produces orbit equivalent flows for all the choices according to Subsection 4.1.

Surgery on projectively Anosov flows
Throughout the rest of the paper, we consider projectively Anosov flows and supporting bi-contact structures as equivalent objects.
In [Sal21] we introduced a Dehn type surgery along a Legendrian-transverse knot K in a 3-manifold M endowed with a bi-contact structure (ξ, η), satisfying certain requirements near K. The procedure can be thought as first cutting the manifold M along an annulus A and then gluing back the two sides of the cut adding a Dehn twist. What is mostly peculiar about this construction is that the surgery annulus is tangent to the vector field X defined by (ξ, η). This choice induces a 3-dimensional deformation of the flow near the knot that has some advantageous dynamical consequence when we work in the contact category.
In this section we introduce an analogous construction using a transverse annulus C. Working on a transverse surface implies remarkable advantages when we consider a general projectively Anosov flow. For instance, we can always chose the supporting bi-contact structure in such a way that our surgery yields always a smooth plane field.
The strategy for defining the surgery is similar to the one used in [Sal21]: we first find a suitable coordinate system where two contact forms (α, β) defining the bi-contact structure take a simple form. From now on, ξ denotes the contact structure containing T K while η denote the transverse one (see Figure 5).  induced on C by the following pair of contact forms (α, β) supporting (ξ, η) Here t ∈ (−τ, τ ) is the parameter given by the flow of X.
Proof. Let (α 1 , β 1 ) be a pair of contact forms supporting X and such that T K ⊂ ker α 1 . Consider the parameter u defined by the flow φ u of the Reeb vector field R α 1 of α 1 . The transverse surgery annulus C is defined as follows. We start with the Legendrian-transverse knot K : S 1 → M and we set with ν ∈ R + . The characteristic bi-foliation induced by the bi-contact structure (ξ = ker α 1 , η = ker β 1 ) is composed by two families of transverse curves on C. This induces a coordinate system (w, s) where the w-curves, induced by the characteristic foliation of η, are transverse to ∂C, while the s-curves induced by ξ are parallel to ∂C. The transverse parameter t is determined by the flow φ t of X and we set t = 0 on C. Note that the s-curves are Legendrian for ξ on C while they are transverse to the contact planes everywhere else in a neighborhood of C. The w-curves are instead everywhere transverse to ξ in a neighborhood of C, while the t-curves are Legendrian curves by definition. This implies that in a neighborhood of C there is a contact form α 2 defining ξ such that Similarly, the s-curves are everywhere transverse to η in a neighborhood of C, while the w-curves are Legendrian for η on C and transverse everywhere else in a neighborhood of C. Since the t-curves are Legendrian by definition there is some contact form β 2 defining η such that β 2 = ds + b 2 (w, s, t) dw, β 2 = ds on C. Sliding the contact planes of ξ and η independently and smoothly along the t-curves induces isotopies of the contact structures that do not change the flow-lines. Therefore in a neighborhood Λ τ of C there is a pair of contact forms (α, β) supporting (ξ, η) such that α = dw + a(t) ds, β = ds + b(t) dw with t ∈ (−τ, τ ), with a(0) = 0 and b(0) = 0. Figure 6. The shear F does not preserve the contact form β. We introduce perturbations in order to produce a smooth plane field after the application of the shear.

Definition of the shear and the perturbations.
We define the shear along the surgery annulus C similarly to [FH13], [FHV19] and [Sal21]. Consider where q ∈ Z and g : R → [0, 2π] is a non-decreasing smooth function such that g((−∞, −1)) = 0 and g([1, ∞)) = 2π and 0 ≤ g ≤ 4. The shear defines a smooth 3-manifoldM diffeomorphic to a one obtained by (1, q)-Dehn surgery, with respect to the contact framing induced by ξ. Since on a neighborhood of C we have α is not preserved by the shear. We perturb α to a 1-formα defined onM \ C in such a way that the plane field distribution kerβ stays transverse to ker α and defines a smooth contact structure after applying the shear. To this end we setα ± = α ± ρ with where λ 1 : R → [0, 1] is a smooth bump function with support in a neighborhood of C defined by t ∈ [−τ 1 , τ 1 ] and such that λ(0) = 1. With this choiceα is a 1-form defined on the whole new manifoldM as the following computation shows Similarly, β is not preserved by the shear since We setβ ± = β ± σ σ = 1 2 λ 2 (t)f (w) dw Figure 7. Although F does not preserve the contact form α in a neighborhood, it does preserve it on the surgery annulus C. As a consequence, in a sufficiently small neighborhood of C, for every q ∈ Z, there is a perturbation of α such that the surgery produces a smooth contact formα.

Theorem 5.3. Given a Legendrian-transverse knot in a bi-contact structure there is an infinite family of Dehn type surgeries that yield projectively Anosov flows in the new manifold.
Proof. The 1-formα is a contact form if and only if If we deform the contact structure along the t-curves in such a way that a(t) = t in a neighborhood of C described by t ∈ [−τ 1 , τ 1 ] the condition becomes where, similarly to [FH13] we chose λ 1 : R → [0, 1] smooth bump function with support in [−τ 1 , τ 1 ] such that |λ 1 | ≤ π/τ 1 . If we chose The contact condition onβ translates in the following inequality Since λ 2 (t) ≥ 0 for t < 0 and λ 2 (t) ≤ 0 for t > 0, choosing τ 2 > τ 1 both the contact conditions are verified if q ∈ N and η is a positive contact structure (therefore ξ is negative) end we have a new bi-contact structure. Obvously if η is a negative contact structure the contact conditions are always verified for −q ∈ N.

Dynamical interpretation of the surgery
Hozoori shows in [Hoz21a] that if φ t is a C 1 -Hölder Anosov flow with orientable weak invariant foliations, there is a special pair of supporting contact forms (α, β) such that, in a neighborhood of a periodic orbit γ, the Reeb vector field R α of α belongs to ker β. This implies that a closed orbit γ can be pushed to a Legendrian-transverse knot K using the flow of the vector field R α . Indeed, since R α is Legendrian for ker β, the contact plane rotate all in the same direction along the flowlines of R α , while the planes of ker α stay invariant (Figure 8). This answer positively to a question posed in [Sal21].
Corollary 6.1 (Hozoori [Hoz21a]). Let φ t be a C 1 -Hölder Anosov flow on a 3-manifold with orientable weak invariant foliations. There is a supporting bi-contact structure such a neighborhood of a periodic orbit γ contains an isotopic Legendrian-transverse knot K.
We call a knot K obtained by flowing a closed orbit γ to a Legendrian-transverse knot a Legendrian-transverse push-off of γ. The above result allows us to construct the transverse annulus C and to apply the surgery as in lemma Lemma 5.1 and Subsection 5.1. We now show that, when the flow is Anosov, there is a strong relation between our bi-contact surgery and Goodman's operation. Proof. The perturbation of the bi-contact structure defined in Subsection 5.1 does not change the flow lines. Therefore for the same choice of the shear and the surgery annulus, the bi-contact surgery reconstructs the same flow lines of Goodman's construction and vice versa (see Figure 4 and Figure 5).

Anosovity of the new flow and applications
It was only few years after the publication of Goodman's work [Goo83] that experts realized that a version of Goodman surgery can be performed along a knot transverse to both the stable and unstable foliations, and the proximity of a closed orbit is not needed [Bar06]. This naturally translates to our setting since a Legendrian-transverse knot is automatically transverse to both the stable and unstable foliations. The use of the cone criterion shows that positive surgeries yield always Anosov flows. However, it is not obvious which is the sequence of negative surgeries that do not break the hyperbolicity condition. We give a contact geometric version of the cone criterion that shows how the Anosovity of the new flows generated by Goodman surgery boils down to contact geometry.

Theorem 7.1. Let M be a 3-manifold endowed with an Anosov flow. The surgery of Subsection 5.1 yields an Anosov flow if and only if it yields a projectively Anosov flow.
Proof. We use the new bi-contact structure to construct two Lorentzian metricsQ ± for the new vector fieldX. We want to show that the metrics satisfy Proposition 2.5. Note that, after surgery, the curve defined by w = 0 it is still a Legendrian-transverse knot in the new bi-contact structure, therefore we can apply Lemma 5.1 and find in Λ τ two special contact formsα = dw +ã ds,ã(0) = 0, β = ds +b dw,b(0) = 0. defining the new flow in Λ τ . Without loss of generality we assume thatα defines a negative contact structure whileβ defines a positive one so that bothã andb are positive. We define the new metricsQ ± as follows. Since the initial vector field X is Anosov there are two continuous Lorentz metrics Q ± as in Proposition 2.5. We use partitions of unity to arrange the new metricsQ ± to have the form in the flow box neighborhood Λ τ and to coincide with Q ± outside Λ τ . Here c is a small enough positive constant to ensure that the positiveQ ± -cone contains E ± in Λ τ . We set In the flow-box Λ τ the flow is represented by a shift in time t 0 + t, therefore we get We show that condition 1 of Proposition 2.5 is verified. To this end it is enough to verify thatQ ± (D xφ ±t ( w)) is strictly increasing in t ∈ [−τ, τ ]. Combined with the exponential growth outside Λ τ implies condition 1 of Proposition 2.5. We compute the derivative Figure 9. The bi-contact structure is represented by the red and blue planes. The white plane is ker dt (the ws-plane).
Note that if We claim that this condition is always achieved if the pair (α,β) defines a bi-contact structure. After an isotopy along the flowlines inside Λ τ we can always write for t ∈ [−τ, τ ]. With this assumption inequality (2) is satisfied if and only if Note that this is exactly the condition of transversality for ker α and ker β in the flow box Λ τ . Condition 2 of Proposition 2.5 is obvious, while a direct computation shows thatQ ± (X) = ±α(X)β(X) − c dt 2 (X) = −c.
We prove condition 4 of Proposition 2.5. We just need to show that for every x ∈ Λ τ we have D xφ ±t (C ± (x)) \ {0} ⊂ C ± (φ ±t (x)) for t > 0. To this end it is enough to observe that the interior of C ± (φ ±t (x)) is given by the inequalitỹ Figure 10. The surgery annulus is the shaded region on the right, while the oblique red line is the image B(γ) of γ (direction perpendicular to K) under the monodromy. The blue curve l is a leaf of the characteristic foliation induced by ξ on a smooth surface S ⊂ Λ τ .
Remark 7.2. The main difference between our metrics and the ones defined in [FH13] and [Bar06] is that in our construction the cones contract for t > 0. In particular the differential of the flow is not aQ ± -isometry in Λ τ .
We give an application of corollary to the following class of knots not isotopic to any closed orbit in a suspension flow. Example 7.3. Consider a T 2 -bundle with Anosov linear monodromy equipped with a propeller (ξ, η) defining the suspension flow, with η positive. When the characteristic foliation induced by ξ on the toric fiber Σ z have rational slope, a leaf is a knot K that is Legendrian for the negative contact structure ξ and transverse for η. Note that K is not isotopic to any closed orbit of the flow since these have non-zero intersection with the fiber (see Figure 10). Proof. Consider a T 2 -bundle with Anosov linear monodromy equipped with a propeller (ker α = ξ, ker β = η) defining the suspension flow and suppose η positive. Call Σ 0 the torus leaf containing K. Without loss of generality we assume that the Legendrian-transverse knot K is one of the generators of the homology of the Σ 0 . Moreover we assume that the characteristic foliation induced by η on Σ 0 has circle leaves and we call γ the corresponding homology class. The manifold C ×[0, τ ] where C = K ×(− , ) ⊂ Σ 0 is a flow-box containing K. By Theorem 7.1 we have just to find perturbations ρ and σ of the original bi-contact structure that define a new bi-contact structure after surgery. We use perturbations with support in [0, τ ] as in remark Remark 5.2. By the proof of Theorem 5.3 we know that there is a prturbation ρ such thatα = α + ρ is a contact form after any (1, q)-surgery with q ∈ Z. We just need to check that there exists a perturbation σ as descried in Subsection 5.1 such Figure 11. On the left, the Legendrian-transverse knot and the positive contact structure before the application of the perturbation. If the slope of B(γ) is grater than the absolute value of the slope of the surgery q/2 , the funcion f defining the shear can be constructed projecting into the surgery annulus the leaves of the characteristic foliations induced by η on a smooth surface S ⊂ Λ τ .
that the 1-formβ = β + σ defines a contact structure. The perturbation exists if and only if the slope of B(γ), image of γ under the monodromy B, is greater than the absolute value of the slope of the surgery q/2 (see Figure 10 and Figure 11).
Example 7.5. Consider the T 2 -bundle over the circle with Anosov linear map B : T 2 → T 2 with B = 5 9 1 2 .
Chose a knot K as in Example 7.3 with positive preferred direction. Since the slope of the image of the second generator of the homology is 9/2, Goodman surgery yields Anosov flows for negative surgery coefficients if q 2 < 9 2 .
In particular, since the upper bound of the width 2 of the surgery annulus is 1, the maximal sequence of surgery coefficients that yield Anosov flow is {q ∈ Z, q ≥ −4}.

Interpretation as contact-transverse surgery and consequences
The formulation of Goodman surgery in terms of bi-contact structures renders immediately obvious the relationships between Goodman's construction and the various flavours of classic contact surgery operation. The bi-contact (1, q)-surgery of definition Subsection 5.1 acts as a contact (1, q)-surgery on ξ. Suppose that, without loss of generality, our construction yields a new bi-contact structure (ξ,η) withη positive. If q < 0 the transverse surgery can be interpreted as an admissible transverse 1/q-surgery (it removes twisting) while if q > 0 as an inadmissible transverse surgery (it adds twisting). We remark that the surgery coefficient q is with respect to the contact framing induced by ξ.
Since the supporting bi-contact structures of an Anosov flow are tight (hypertight as shown by Hozoori in [Hoz20b]), we can use corollary Theorem 7.1 to deduce some (hyper)tightness result of classic surgery operation in contact geometry. As usual, let q be an integer.
Corollary 8.1. Let K be a Legendrian-transverse knot in a bi-contact structure (ξ, η) defining an Anosov flow and suppose η positive. Every inadmissible transverse 1/q-surgery on η along K yields a hypertight contact structureη.
Proof. Every inadmissible transverse 1/q-surgery on η along K can be realized by a bicontact 1/q-surgery with coefficient q > 0. This yields a bi-contact structure and hence an Anosov flow by Theorem 7.1, thusη is hypertight.
The following result of Conway allows us to translate Corollary 8.1 in terms of positive contact surgeries.

Theorem 8.2 ([Con19]). Every inadmissible transverse surgery on a transverse knot corresponds to a positive contact surgery on a Legendrian approximation.
Suppose that γ is a periodic orbit of an Anosov flow and consider a supporting bi-contact structure (ξ = ker α, η = ker β) such that R α ∈ ker β and suppose η positive. Consider the transverse push-off of γ along R α . Such a Legendrian-transverse knot can be pushed to the original closed orbit by a Legendrian push-off using the flow of −R α . The closed orbit is the Legendrian approximation of Theorem 8.2 and we have the following. Corollary 8.3. Let γ be a closed orbit in an Anosov flow and let (ξ = ker α, η = ker β) a supporting bi-contact structure such that R α ∈ ker β and suppose η positive. A positive 1/q-contact surgery along γ on η yields a new hypertight contact structure.
Proof. Push the closed orbit γ to a Legendrian-transverse knot using the flow of R α . If q > 0 the bi-contact surgery induces an inadmissible transverse surgery along K on η by construction. Corollary 8.1 implies that the new contact structureη is hypertight and Theorem 8.2 ensures that the same contact structureη can be realized (up to isotopy) as a contact surgerey along the Legendrian approximation of K given by γ.
Example 8.4. As observed in the introduction, a consequence of Corollary 5 is that a positive 1/q-contact surgery on any suspension knot of a T 2 -bundle Anosov monodromy equipped with a vertically rotating positive contact structure with minimal twisting yields an hypertight contact structure.

Non-Anosov bi-contact structures on hyperbolic 3-manifolds
We finally apply our construction to produce families of bi-contact structures that do not define Anosov flows. Our examples are generated applying the bi-contact surgery on suitable propellers as described by Mitsumatsu ([Mit95], [Mit02], see Theorem 5.3). In particular, we show how to generate new non-Anosov bi-contact structures on the family of 3-manifolds {M (q), q ∈ N} as described in the introduction. We remark that M (q) is hyperbolic except when q ∈ {0, 1, 2, 3, 4}.
Consider on M (0) equipped with the pair of vertically rotating tight contact structures (ξ = ker α, η n = ker β n ) as described in Example 2.4. More precisely α defines a negative, vertically rotating contact structure with rotation angle θ 0 less then π while β n defines a positive, vertically rotating contact structure with rotation 2nπ < θ n < π(2n + 1). We Figure 12. The negative contact structure ker β n (in red) has non-minimal twisting. The positive one ker α (in blue) defines the suspension flow. The framing of the surgery annulus associated to a suspension knot K is determined by ker α. Clearly K is Legendrian transverse for the perturbed plane field defined by the pair of contact forms (α, β n + dz).
consider the family of bi-contact structures (ξ, η n ) defined by the pairs of contact forms (α, β n + dz) where ∈ R.
Proof. Clearly K is a Legendrian-transverse knot for (ξ, η n ) (see Figure 1), therefore if q ∈ N a bi-contact 1/q-surgery yields a bi-contact structure. Since it acts as a contact 1/q-surgery on ξ, the new contact structureξ is hypertight. Since the framing of the surgery annulus is induced by the contact framing of ξ, the new manifold is M (q).
Corollary 9.2. The weak invariant foliations F u and F s of the flow defined by the bicontact structure as above contain leaves of genus g > 0, therefore the flow is projectively Anosov but not Anosov. Figure 13. On the left, a compact leaf Σ of the original invariant foliation. The shaded disk represents the intersection with the neighborhood Λ of the suspension knot K. On the right, the leaf Σ after surgery.
Proof. Since the rotation of the contact planes of η n along the z-axis is θ n > 2π there is a torus leaf Σ in one of the invariant foliations corresponding to the z where the plane fields ker α and ker β n coincide (see Figure 1). Note that our operation perturbs the bi-contact structure just in a neighborhood Λ of the Legendrian-transverse knot K, and K is transverse to Σ. Therefore, the new invariant foliationsF u andF s contain a leafΣ such that (Σ \ D) ⊂Σ where D = Σ ∩ Λ. Since g(Σ \ D) > 0 it follows that g(Σ) > 0 (see Figure 13).