Daniel Mathews

This article is the fifth in a series on Liouville and contact geometry, on convex surfaces and characcteristic foliations.

In our previous episode, we saw that when you have a product neighbourhood \(s \times[-1,1]\) of a surface \(S\) in a contact 3-manifold, you get a family, or “movie”, of characteristic foliations \(\mathcal{F}_t\) on the surfaces \(S_t = S \times {t}\). When \(S\) is convex and the neighbourhood \(S \times [-1,1]\) is defined from a transverse contact vector field, the foliations \(\mathcal{F}_t\) are all the same, \(\mathcal{F}_t = \mathcal{F}\).

We then asked the reverse question: if you have a family of foliations \(\mathcal{F}_t\) on the surface \(S\), do they arise as the movie of characteristic foliations of a contact structure on \(S \times [-1,1]\), i.e. with \(\mathcal{F}_t\) being the characteristic foliation on \(S_t\)? And we saw a couple answers. Under certain circumstances, a movie of foliations \(\mathcal{F}_t\) is the movie of a contact structure \(\xi\) — and depending on what you know about the \(\mathcal{F}_t\), you might know something about \(\xi\).

In this episode, we ask the question of how unique these contact structures are. If you have two contact structures \(\xi, \xi’\) with the same movie of foliations \(\mathcal{F}_t\), are they the same, or equivalent in any sense? And is it possible to have two movies of foliations \(\mathcal{F}_t, \mathcal{F}’_t\) which are the movies of equivalent contact structures?

Before attacking these questions, let’s recall what we saw previously, and let’s figure out what we might mean by “equivalence of contact structures”.

Recall we’ve said that a foliation \(\mathcal{F}\) on a surface \(S\) divides \(S\) if there is a curve (dividing set) \(\Gamma\) which cuts \(S\) into two pieces \(S_+, S_-\) on which \(\mathcal{F}\) is directed by a vector field which expands an area form. In this case you get a Liouville structure on each of \(S_+\) and \(S_-\). This is precisely the sort of foliation we see on a convex surface in a contact 3-manifold.

We saw that if all foliations \(\mathcal{F}_t\) are the same, and divide \(S\), then they are the movie of a contact structure on \(S \times [-1,1]\) — and in fact the contact structure is invariant in the \(t\) direction, and makes \(S\) convex.

We also saw Giroux’s realisation lemma, which says that if each foliation \(\mathcal{F}_t\) divides \(S\), then again, the foliations form the movie of a contact structure on \(S \times [-1,1]\).

This is all very nice. Our agenda, however, is to understand to what extent these contact structures are unique, or equivalent. So let’s examine two possible versions of equivalence of contact structures.

One standard way to consider two contact structures “equivalent” is if they are isotopic. A contact structure on a manifold \(M\) is a type of plane field on \(M\). If you can just continuously deform one into the other, they should be equivalent — but we need to be a little bit careful, because contact structures are plane fields with a special condition, that of non-integrability.

A homotopy of plane fields is a continuously varying family of plane fields, so two plane fields are homotopic if you can turn one into the other by a continuous deformation. An isotopy of contact structures is an homotopy of plane fields, where the plane field at each instant of time is in fact a contact structure.

So, two contact structures are isotopic if you can turn one into the other via a continuous deformation of the contact planes, but the planes must always retain the contact property of non-integrability.

Another way to consider two contact structures “equivalent” is if they are

related by a diffeomorphism of the manifold \(M\). A diffeomorphism \(\phi : M \rightarrow M\) has a derivative \(\phi_*\) which sends tangent vectors to tangent vectors, tangent planes to tangent planes, and contact structures to contact structures. Two contact structures \(\xi, \eta\) are are related by a diffeomorphism \(\phi\) if \(\eta = \phi_* \xi\).

For now, I’m more interested in continuous deformations of contact structures, rather than diffeomorphisms. But you can also obtain continuous deformations of contact structures from such diffeomorphisms!

Rather than considering a single diffeomorphism \(\phi\) of \(M\), we can have a 1-parameter family of diffeomorphisms \(\phi_t\), where each \(\phi_t\) is a diffeomorphism \(M \rightarrow M\). The \(\phi_t\) vary continuously in \(t\), say for \(t \in [0,1]\). And since we want to start from where we’re at, we usually require \(\phi_0\) to be the identity. This idea is sometimes called a diffeotopy. We can think of a diffeotopy as an ambient isotopy — the points of the whole 3-dimensional space \(M\) are moved about!

Such ambient isotopies naturally arise as the flows of vector fields. When you have a vector field \(X\) on a manifolld \(M\), if you flow along \(X\) for time \(t\) you obtain a diffeomorphism \(\phi_t : M \rightarrow M\). The family of diffeomorphisms \(\phi_t\) is a continuously varying family of diffeomorphisms of \(M\), starting from \(\phi_0\), which is the identity.

So this gives us two distinct notions of “continuous deformation” of a contact structure.

• Isotopy of contact structures: A family of contact structures \(\xi_t\) on \(M\) which varies continuously. In other words, the contact planes move from one contact structure to another, through contact structures.
• Ambient isotopy of contact structures: Given family of diffeomorphisms \(\phi_t : M \rightarrow M\), which varies continuously from \(\phi_0 = \text{Identity}\), starting from the contact structure \(\xi = \xi_0\) we obtain a family of contact structures \(\xi_t = \phi_{t*} \xi\). In other words, the whole space moves, and carries the contact plane field along with it!

Now it’s hopefully clear that an ambient isotopy induces an isotopy of contact structures. But it’s not at all clear that an isotopy of contact structures should arise from an ambient isotopy.

But although it’s not at all clear, it’s true! These two notions of “continuous deformation of contact structures” are essentially the same! This is known as Gray’s theorem.

GRAY’S THEOREM: Let \(\xi_t\) for \(t \in [0,1]\) be an isotopy of contact structures on a compact 3-manifold \(M\) without boundary. Then there exists a family of diffeomorphisms \(\phi_t : M \rightarrow M\), for \(t \in [0,1]\), such that that \(\phi_{t*} \xi_0 = \xi_t\).

Gray’s theorem is even more amazing, because the proof is explicit! It tells you how to find the diffeomorphisms \(\phi_t\). The method of this proof, sometimes known as Moser’s method, constructs \(\phi_t\) as the flow of a vector field \(X\), and uses the properties of symplectic and contact structures to find that vector field.

This, however, leads to a problem when \(M\) has boundary. The statement of the theorem applies only when \(M\) has no boundary. The method works in general, but the problem is that when \(M\) has boundary, the vector field \(X\) will in general point in or out of the boundary. Thus you can’t necessarily define the flow \(\phi_t\), as you might flow out of the manifold — there be dragons!

And, if we are just considering a neighbourhood \(S \times [-1,1]\) of a surface \(S\), then this is an issue, because \(S \times [-1,1]\) very definitely has boundary, namely at \(S \times {-1,1}\) !

Anyway, let’s return to our first question: if you have two contact structures \(\xi, \xi’\) with the same movie of foliations \(\mathcal{F}_t\), are they the same, or equivalent in any sense? Giroux showed that they are: they are then isotopic. He called this his “reconstruction lemma”; it’s lemma 2.1 of “Structures de contact en dimension trois et bifurcations des feuilletages de surfaces”.

RECONSTRUCTION LEMMA: If two contact structures on \(S \times [-1,1]\) have the same characteristic foliations \(\mathcal{F}_t\) on each surface \(S_t = S \times {t}\), then they are isotopic.

In other words, if two contact structures have the same movie, they are isotopic.

(Giroux says further that the two contact structures are isotopic relative to the boundary, but I don’t believe it, at least not in the sense of what I understand it to mean. The two contact structures could be quite different on \(S \times {-1,1}\), and hence the isotopy connecting them must be nontrivial on the boundary. But perhaps Giroux means something else.)

Geometrically, if two contact structures \(\xi, \xi’\) have the same movie, then on each surface \(S_t\), they draw the same characteristic foliation \(\mathcal{F}_t\), so they are both tangent to the lines of \(\mathcal{F}_t\). The contact planes of \(\xi, \ xi’\) just spin around those lines differently!

The proof of the reconstruction lemma is not very difficult in the end. It relies upon a calculation we saw before . Namely, a contact form can be written as \(\alpha = \beta_t + u_t \; dt\), where \(\beta_t\) is a 1-form whose kernel on \(S_t\) yields \(\mathcal{F}_t\), and \(u_t\) is a real-valued function on \(S_t\). We saw that the condition for \(\alpha\) to be a contact form is that
\[
u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t )
\]
be an area form on each \(S_t\). Fixing an orientation on \(S_t\), we can write this requirement as the inequality
\[
u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t ) > 0.
\]

Now, our two contact structures \(\xi,\xi’\) draw the same movies, which are the characteristic foliations \(\mathcal{F}_t\), which are given by the kernels of \(\beta_t\). So we can take contact forms \(\alpha, \alpha’\) which have the same \(\beta_t\) terms.

The key idea is to take the inequality above, with fixed \(\beta_t\), and consider the set of all \(u_t\) which would satisfy the inequality. The key observation is that this is a convex set. For if \(u_t, v_t\) are two functions which satisfy the inequality, then so too does any convex linear combination \((1 – \lambda) u_t + \lambda v_t\), for any \(\lambda \in [0,1]\).

Explicitly, if we have
\[
u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t ) > 0
\quad \text{and} \quad
v_t \; d\beta_t + \beta_t \wedge ( dv_t – \dot{\beta}_t ) > 0.,
\]
then taking \((1 – \lambda)\) times the first inequality plus \(\lambda\) times the second, since both \(\lambda, 1 – \lambda \geq 0\), yields
\[
[ (1- \lambda) u_t + \lambda v_t ] \; d\beta_t + \beta_t \wedge ( d [ (1 – \lambda) u_t + \lambda v_t ] – \dot{\beta}_t ) > 0,
\]
so that replacing \(u_t\) with \((1 – \lambda) u_t + \lambda v_t\) in the original inequality, the inequality still holds.

With this observation in hand, it’s not difficult to prove the lemma.

PROOF OF LEMMA. Let the two contact structures be \(\xi_0, \xi_1\). As they have the same movie, these two contact structures have contact forms \(\alpha_0 = \beta_t + u_t \; dt\), \(\alpha_1 = \beta_t + v_t \; dt\), where \(\beta_t\) is a 1-form and \(u_t, v_t\) are real-valued functions. We can take the same \(\beta_t\) in both contact forms precisely because they have the same movie of foliations.

Now the contact conditions for \(\alpha_0, \alpha_1\) are precisely given by the two inequalities above for \(u_t\) and \(v_t\). For \(\lambda \in [0,1]\), define a 1-form \(\alpha_\lambda\) as a convex linear combination of \(\alpha_0\) and \(\alpha_1\):
\[
\alpha_\lambda = (1-\lambda) \alpha_0 + \lambda \alpha_1
= \beta_t + [ (1-\lambda) u_t + \lambda V_t ] \; dt
\]
Now as discussed above, \(u_t, v_t\) satisfy the desired inequalities, and hence so too does \((1-\lambda) u_t + \lambda v_t\). And this convex linear combination satisfying the inequality means that \(\alpha_\lambda\) is a contact form. So we have a continuously varying family of contact forms \(\alpha_\lambda\), from \(\alpha_0\) to \(\alpha_1\). This gives an isotopy of contact structures from \(\xi_0\) to \(\xi_1\). QED

This lemma gives a very nice answer to our first question. Yes, it says, if two contact structures give the same movie of foliations, then they are equivalent — they are isotopic.

But what about ambient isotopy? Well, as mentioned above, Gray’s theorem construct a vector field whose flow will give an ambient isotopy — the problem is that this vector field might point in or out of the boundary of \(S \times [-1,1]\). If we’re happy to have our diffeomorphisms going beyond \(S \times [-1,1]\), there’s no problem. But if we want to stay with everything happening in \(S \times [-1,1]\), we may have a problem.

In any case, let’s now turn to our second question; and in fact in answer to this question we will be able to give an answer involving ambient isotopy. The question we asked was: Is it possible to have two movies of foliations, which are movies of equivalent contact structures?

It’s certainly possible. In fact, we can construct such a situation to involve an ambient isotopy.

Let’s start from our old, classic, convex surface situation. Let’s consider a contact structure \(\xi\) near a convex surface \(S\), with a neighbourhood \(S \times [-1,1]\) defined by a transverse contact vector field, so that all the foliations \(\mathcal{F}_t\) are the same, i.e. the movie of foliations is all just the same frame. In this case the contact structure is “vertically invariant” and we have a contact form \(\alpha = \beta + u \; dt\), where \(\beta, u\) are a 1-form and a real-valued function on \(S\), with no dependence on \(t\).

Now, let’s consider a diffeomorphism \(\phi\) of \(S\). In fact, let’s consider a smooth family of diffeomorphisms \(\phi_t\) of \(S\), starting from \(\phi_0\) being the identity, through to \(\phi_1\) being our diffeomorphism \(\phi\). So \(\phi\) is a diffeomorphism which is isotopic to the identity, and \(\phi_t\) is an isotopy of diffeomorphisms, or diffeotopy. So for each \(t \in [0,1]\), we have a diffeomorphism \(\phi_t : S \rightarrow S\), and these vary smoothly in \(t\), with \(\phi_0\) the identity, and \(\phi_1\) being our original diffeomorphism \(\phi\).

Let’s now use the diffeomorphisms \(\phi_t\), over all \(t \in [0,1]\), to construct a diffeomorphism \(\Phi\) of \(S \times [0,1]\). We’ll define \(\Phi\) by applying \(\phi_t\) to each surface \(S_t = S \times {t}\). In other words,
\[
\Phi(x,t) = (\phi_t (x), t).
\]
Note that we’ve only taken \(t \in [0,1]\) here, but we have a larger interval \([-1,1]\) in our thickened surface \(S \times [-1,1]\). But since \(\phi_0\) is the identity on \(S\), we can extend \(\Phi\) to in fact be a diffeomorphism of the whole \(S \times [-1,1]\) by being the identity on \(S \times [-1,0]\).

Thus we have a diffeomorphism \(\Phi : S \times [-1,1] \rightarrow S \times [-1,1]\). It preserves each slice \(S_t\). It’s the identity on each \(S_t\), for \(t \leq 0\). But for \(t \geq 0\) it moves the slices about in a smooth fashion, starting from the identity on \(S_0\), through to applying \(\phi\) to \(S_1\).

Applying the diffeomorphism \(\Phi\) (or rather, its derivative) to the nice original vertically invariant contact structgure \(\xi\), we obtain another contact structure. Let \(\eta= \Phi_* \xi\).

So \(\eta\) is another contact structure on \(S \times [-1,1]\). It’s related to \(\xi\) by the diffeomorphism \(\Phi\). Now since \(\Phi\) preserves each surface \(S_t\) and \(\Phi\) also sends the contact planes of \(\xi\) to \(\eta\), it must send the characteristic foliations of \(\xi\) to the characteristic foliations of \(\eta\). If we define \(\mathcal{G}_t\) to be the characteristic foliation of \(\eta\) on \(S_t\), then \(\Phi (\mathcal{F}_t) = \mathcal{G}_t\). Indeed, thinking purely about the individual slice \(S_t\), we have \(\phi_t (\mathcal{F}_t) = \mathcal{G}_t\). So \(\mathcal{F}_t\) is the movie of foliations of \(\xi\), and \(\mathcal{G}_t\) is the movie of foliations of \(\eta\).

Now recall the original contact structure \(\xi\) was vertically invariant, so all the foliations \(\mathcal{F}_t\) are the same, say \(\mathcal{F}_t = \mathcal{F}\). But \(\phi\), on the other hand, can be any diffeomorphism of \(S\) isotopic to the identity — deforming the points of \(S\) around in some fashion. So the movie of foliations \(\mathcal{G}_t = \phi_t (\mathcal{F}_t) = \phi_t (\mathcal{F})\) will in general be very different from \(\mathcal{F}\). In other words, \(\xi\) and \(\eta\) will in general have very different movies of foliations.

And yet, despite \(\xi\) and \(\eta\) having very different movies, they are related to each other by the diffeomorphism \(\Phi\). We claim that they are in fact isotopic — in fact, ambient isotopic.

To show \(\xi, \eta\) are ambient isotopic, we just need to show that the diffeomorphism \(\Phi\) of \(S \times [-1,1]\) is isotopic to the identity. This is not so difficult, since \(\Phi\) is constructed out of the diffeomorphism \(\phi_t\) of \(S\), which are themselves an isotopy from the identity! We just need to straighten out what we mean.

To define the isotopy from \(\Phi\) to the identity on \(S \times [-1,1]\), we need a new time variable! We’ve already used \(t\) for the coordinate on \([-1,1]\). So let’s use a new variable \(s\). We’ll define a family of diffeomorphisms \(\Phi_s : S \times [-1,1] \rightarrow S \times [-1,1]\), for \(s \in [0,1]\) varying smoothly in \(s\), with \(\Phi_0\) beint the identity and \(\Phi_1 = \Phi\).

On \(S \times [0,1]\), we defined \(\Phi (x,t) = (\phi_t (x), t)\). We can now define \(\Phi_s\) on \(S \times [0,1]\) by
\[
\Phi_s (x,t) = (\phi_{st} (x), t).
\]
Here \(st\) just means \(s\) times \(t\)! At time \(s\), \(\Phi_s\) acts on the slice \(S_t\) via the diffeomorphism \(\phi_{st}\).

From this definition we clearly have that each \(\Phi_s\) is a diffeomorphism of \(S \times [-1,1]\), and that these diffeomorphisms vary smoothly in \(s\). When \(s = 0\), we have \(\Phi_0 (x,t) = (\phi_0 (x), t)\), and since \(\phi_0\) is the identity on \(S\), this means \(\Phi_0\) is the identity on \(S \times [0,1]\). When \(s=1\), we have \(\Phi_1 (x,t) = (\phi_t (x), t) = \Phi(x,t)\), so \(\Phi_1 = \Phi\). So indeed \(\Phi_s\) is an isotopy of diffeomorphisms of \(S \times [0,1]\) from the identity to \(\Phi\).

Now on \(S \times [-1,0]\), \(\Phi\) is the identity. So there is a clear isotopy from the identity to \(\Phi\) — namely the isotopy just consisting of the identity! So we can extend \(\Phi_s\) to be defined on all of \(S \times [-1,1]\), by letting \(\Phi_s\) be the identity on \(S \times [-1,0]\).

Thus we obtain an isotopy of diffeomorphisms \(\Phi_s\) of \(S \times [-1,1]\) from the identity to \(\Phi\). So if we let \(\xi_s = \Phi_{s*} \xi\), then each \(\xi_s\) is a contact structure, with \(\xi_0 = \xi\) and \(\xi_1 = \eta\). So \(\xi, \eta\) are isotopic, and the diffeotopy \(\Phi_s\) shows that they are in fact ambient isotopic.

Thus, we have constructed examples of contact structures on \(S \times [-1,1]\) which are isotopic, indeed ambient isotopic, but which induce different movies of foliations on the surfaces \(S_t\).

We can summarise this construction in the following proposition.

PROPOSITION: Let \(S\) be a closed surface, and \(\xi\) a vertically invariant contact structure on \(S \times [-1,1]\). Let \(\phi\) be a diffeomorphism of \(S\) isotopic to the identity via an isotopy \(\phi_t\), with \(\phi_t = \phi\) and \(\phi_t\) the identity for \(t \leq 0\). Then the contact structure \(\eta\) defined by
\[
\eta = \Phi_* \xi,
\quad \text{where} \quad
\Phi(x,t) = (\phi_t (x), t)
\]
is ambient isotopic to \(\xi\), via the diffeotopy \(\Phi_s\) of \(S \times [-1,1]\) defined by \(\Phi_s (x,t) = ( \phi_{st} (x), t)\).

This actually has a nice application when we consider adding bypasses.

Bypasses are objects which are attached to the boundary of a contact 3-manifold along a type of arc called an attaching arc. An attaching arc is a special type of arc on a convex surface \(S\). Specifically, an attaching arc must (i) run along the characteristic foliation of \(S\), (ii) begins and ends on the dividing set, and (iii) intersects the dividing set at a single point of its interior. Thus an attaching arc intersects the dividing set in preciselly three points.

We’ll also consider arcs which satisfy only (ii) and (iii). That is, they intersect the dividing set in the same pattern, but they might not run along the characteristic foliation. We’ll call these combinatorial attaching arcs.

We’ll let \(\gamma\) be an attaching arc on the convex surface \(S\) in the contact manifold \(S \times [-1,1]\) with vertically invariant contact structure \(\xi\), and we’ll let \(\delta\) be a combinatorial attaching arc. Moreover, we’ll suppose that there is an isotopy of combinatorial attaching arcs between \(\gamma\) to \(\delta\). In other words, you can slide \(\gamma\) to \(\delta\) along \(S\) through combinatorial attaching arcs.

PROPOSITION: \(\xi\) is ambient isotopic to a contact structure \(\eta\) on \(S \times {1}\) for which \(\delta \times {1}\) is an attaching arc.

Obviously \(\delta \times {1}\) is a combinatorial attaching arc: the point is that we can adjust the contact structure to make it a bona fide attaching arc, running along the characteristic foliation.

PROOF: The isotopy of arcs from \(\gamma\) to \(\delta\) extends to an isotopy of diffeomorphisms of \(S\), which preserves the dividing set \(\Gamma\). In other words, we can obtain a family of diffeomorphisms \(\phi_t : S \rightarrow S\), varying smoothly in \(S\), with \(\phi_0\) being the identity, \(\phi_1 (\gamma) = \delta\), and for all \(t\), \(\phi_t (\Gamma) = \Gamma\).

(This \(\phi_t\) can be constructed by first extending the isotopy from \(\gamma\) to \(\delta\), to an isotopy from \(\gamma \cup \Gamma\) to \(\delta \cup \Gamma\). Then the remaining pieces of the surface can be carried to each other.)

The construction in the previous proposition then provides a diffeomorphism \(\Phi : S \times [-1,1] \rightarrow S \times [-1,1]\), isotopic to the identity via a diffeotopy \(\Phi_s\), and sending \(\xi\) to an ambient isotopic structure \(\eta\). Now \(\Phi(x,1) = \Phi_1 (x,1) = (\phi_1 (x),1)\), so \(\Phi\) acts on \(S_1 = S \times {1}\) via \(\phi_1\). Since \(\Phi\) takes \(\xi\) to \(\eta\), \(\Phi\) sends the characteristic foliation of \(\xi\) to the characteristic foliation of \(\eta\). Since \(\phi_1 (\gamma) = \delta\), this means that \(\delta\) runs along the characteristic foliation of \(\eta\). So \(\delta \times {1}\) is a bona fide attaching arc in the contact structure \(\eta\). QED

This is a form of flexibility of contact structures. We can slide an attaching arc around, and still realise it as an attaching arc, via an ambient isotopy of the contact structure.

It’s known, moreover, that once you give the dividing set, the contact structure nearby is determined, in a certain sense. So what this proposition tells us is that the dividing set and attaching arcs can be considered purely combinatorially, or topologically, on the surface, rather than having to worry too much about characteristic foliations and the detailed geometry of contact planes!

(This article is the fourth in a series on Liouville and contact geoemtry. The first was on Liouville (exact symplectic) geometry on surfaces. The second went from them to convex surfaces in 3-dimensional contact geometry. The third went back, from convex surfaces in 3-dimensional contact geometry, to 2-dimensional Liouville geometry, and showed how convex surfaces can be regarded as two Liouville structures, pieced together along a dividing set.)

We’ve seen that there are excellent things called convex surfaces in 3-dimensional contact geometry, closely related to Liouville geometry. Indeed, on convex surfaces we have wondeful foliations. So when you slice a contact 3-manifold along a convex surface, you get wonderful foliations. We’re now going to consider the relationship between these foliations on surfaces, and contact structures

Recall a convex surface is a surface \(S\) in a 3-dimensional contact manifold \((M, \xi)\) which have a transverse contact vector field \(X\). We’ve seen how you can use the vector field \(X\) to define a transverse coordinate \(t\) and hence describe a neighbourhood of \(S\) as \(S \times [-1,1]\), where \(S = S \times {0}\), with \(t\) giving the latter coordinate. With respect to these coordinates, \(\xi\) has a contact form \(\alpha\) of the form
\[
\alpha = \beta + u \; dt,
\]
where \(\beta\) is a 1-form on \(S\), and \(u\) is a real-valued function on \(S\). (We’re assuming everything is smooth here.)

The neighbourhood \(S \times [-1,1]\) of \(S\) comes as a family of surfaces \(S_t = S \times {t}\), with \(S = S_0\). The contact vector field \(X\) flows \(S_0\) to each \(S_t\), and hence each surface \(S_t\) in this family looks the same with respect to contact geometry. The contact planes are preserved by the flow of \(X\).

Thus, on each surface \(S_t\), we have the same characteristic foliation — where by “same”, we mean the surfaces and their foliations are related by the flow of \(X\). When you map one surface \(S_t\) in this family to another by a flow of \(X\), you’ll also map the characteristic foliation on one surface to the characteristic foliation on the other. Thinking of \(X\) as being “vertical” and \(S\) as “horizontal”, it means that the contact structure is “vertically invariant” — it doesn’t change as you move in the vertical direction, from one surface \(S_t\) of the family to another.

This is a really nice structure to have. Rather than having to think about all the surfaces \(S_t\) near \(S\), you really only have to think about one, because they are all the “same”, in this sense.

The most amazing thing about convex surfaces is Giroux’s proof that almost any embedded surface in a contact 3-manifold is convex. So, for “almost any” embedded surface \(S\), there is a transverse contact vector field, and then you know that you can take \(S\) as part of a family of surfaces \(S_t\) foliating a neighbourhood of \(S\), which are “all the same” in the above sense, and hence you only need to think about one of these surfaces!

What does “almost any” surface mean here? Giroux describes it in terms of a property of the characteristic foliation: if the characteristic foliation is “almost Morse-Smale”, then the surface is convex. And “almost Morse-Smale” is a generic property. In particular, given any embedded surface, after a \(C^\infty\) small perturbation, the surface becomes convex.

(For surfaces with boundary, we often want to preserve a little more structure, and so some further details are required. But we will not pursue them here.)

However, let’s now imagine we took a surface \(S\), and we didn’t know about any transverse contact vector field — i.e. we didn’t know if \(S\) was convex or not. Then, we could still take a neighbourhood of \(S\), of the form \(S \times [-1,1]\), and define \(t\) as the coordinate on the latter factor. Then we would again obtain a family of surfaces \(S_t = S \times {t}\) foliating a neighbourhood of \(S\), with \(S = S_0\). But now the surfaces \(S_t\) could all have different characteristic foliations \(\mathcal{F}_t\).

It would be a mess! That is why knowing \(S\) is convex, or equivalently, having a transverse contact vector field, really helps.

Thinking of \(t\) as time, you can think of the family of foliations \(\mathcal{F}_t\) as a “movie” of foliations. Or, since you are probing the contact structure by considering how it cuts the slices \(S_t\), you can think of it as a form of “tomography” — and this is what Giroux calls it.

In a 2000 paper, Giroux studied these sorts of “movies” or “tomography”. The paper is called “Structures de contact en dimension trois et bifurcations des feuilletages de surfaces”, which translates as “Contact structures in dimenson three and bifurcations of foliations of surfaces”. The French word “feuilletage” is much nicer than the English word “foliation”.

Given a surface \(S\) in a contact manifold \((M, \xi)\) with a product neighbourhood \(S \times [-1,1]\), you obtain a movie of foliations \(\mathcal{F}_t\) on the surfaces \(S_t\). If \(S\) is convex then, by choosing the product neighbourhood right (as above), all \(\mathcal{F}_t\) are the “same” (as discussed above) — the “movie” is just one frame, repeated!

But if all you have are the foliations \(\mathcal{F}_t\), you can’t say much at all. In fact, given a family of foliations \(\mathcal{F}_t\) on \(S_t\), it’s not even clear that they come from a contact structure at all!

So a first question is: which families of foliations arise from contact structures? Equivalently: what movies of foliations describe contact structures. Or: what tomography can you get from slicing a contact structure. This question is not easy.

Giroux however gives some answers. I want to consider one of his results, which he calls the “realisation lemma”.

Here is one possible line of reasoning. We have seen that certain characteristic foliations arise from convex surfaces. A characteristic foliation \(\mathcal{F}\) on a convex surface has a dividing set \(\Gamma\), splitting \(S\) into two pieces \(S_+\) and \(S_-\), and on each piece \(\mathcal{F}\) can be directed by a vector field which expands an area form. We’ll say that such a foliation divides \(S\). It would be very pleasing to see this lovely Liouville geometry on each slice. The nicest case would be if we had the same Liouville geometry on each slice.

Does it follow, in these circumstances, that the foliations come from a contact structure?

In other words, suppose we have \(S \times [-1,1]\), and let \(\mathcal{F}\) be a foliation which divides \(S\). Suppose we have the foliation \(\mathcal{F}_t = \mathcal{F}\) on each slice \(S_t = S \times {t}\). Is this the movie of foliations of a contact structure on \(S \times [-1,1]\)?

Giroux proved that the answer is yes. (This is proposition I.3.4 of his 1991 paper “Convexité en topologie de contact”.) In fact we have already seen the basic idea of the proof. Since \(S_+\) and \(S_-\) have Liouville structures, we may take a 1-form \(\beta\) on \(S_+ \cup S_-\), such that \(d\beta\) is an area form on \(S_+ \cup S_-\), and the dual vector field \(X\) with respect to \(\beta\) directs \(\mathcal{F}\) on \(S_+ \cup S_-\). As we saw previously, \(\beta\) being Liouville is equivalent to \(\beta + dt\) being a contact form on \((S_+ \cup S_-) \times [-1,1]\). So we have a contact structure on \((S_+ \cup S_-) \times [-1,1]\). It remains to extend this over \(\Gamma \times [-1,1]\). And Giroux shows that this can be done. He shows that you can patch them together, to obtain a contact form on \(S \times [-1,1]\) which takes the form \(\beta + u \; dt\), where \(\beta\) is a 1-form on \(S\), and \(u\) is a real-valued function on \(S\).

Indeed, having a contact form of this type, the contact structure obtained on \(S \times [-1,1]\) is not just any old contact structure: it’s invariant in the \([-1,1]\) direction. It’s “vertically invariant”, and so we have a contact vector field in the \([-1,1]\) direction. This direction of course is transverse to \(S\), so we have \(S\) is convex.

Great. So a “constant” movie of foliations, where the foliation \(\mathcal{F}\) divides each slice (i.e. has a dividing set which cuts each slice into pieces on which there is a Liouville structure) is always the movie of a contact structure — and indeed a vertically invariant contact structure exhibiting \(S\) as convex.

But let’s suppose we have a slightly worse situation. Suppose we have different foliations \(\mathcal{F}_t\) appearing on the slices \(S_t\), but each individual foliation \(\mathcal{F}_t\) still divides \(S_t\). In other words, each foliation \(\mathcal{F}_t\) has a dividing set \(\Gamma\) which splits \(S\) into an \(S+\) and \(S_-\), and \(\mathcal{F}_t\) can be directed by a vector field which expands an area form on \(S+\) and \(S_-\). Here \(\Gamma\), \(S_+\) and \(S_-\) might all vary with \(t\), so we should really write something like \(\Gamma_t\), \(S_{t,+}\) and \(S_{t,-}\) to indicate the dependence on \(t\).

In these circumstances, are the foliations \(\mathcal{F}_t\) the movie of a contact structure?

It’s not quite so clear. When you have a foliation which can vary with \(t\), the contact condition becomes more complicated.

When you just have a single foliation, with a dividing set and Liouville structures on either side, then you get a contact form of the type \(\alpha = \beta + u \; dt\), where \(\beta\) is a 1-form and \(u\) a real-valued function on \(S\). The condition for a 1-form to be a contact form is that \(\alpha \wedge d\alpha\) be a volume form, i.e. a non-degenerate 3-form. When \(\alpha = \beta + u \; dt\) we have
\[
\alpha \wedge d\alpha
= (\beta + u \; dt) \wedge (d\beta + du \wedge dt)
= (u \; d\beta + \beta \wedge du ) \wedge dt .
\]
So given that \(\beta\) and \(u\) are purely on S, i.e. have no \(t\)-dependence, the condition for \(\alpha = \beta + u \; dt\) to be a contact form is precisely that \(u \; d\beta + \beta \wedge du\) be an area form on \(S\).

But when you have a family of foliations, even if they each divide \(S\), you would be looking for a contact form of the type \(\alpha = \beta + u \; dt\) again — but now \(\beta\) and \(u\) can depend on \(t\). Perhaps it’s better to write, as Giroux does, \(\beta_t\) and \(u_t\), to indicate the dependence on \(t\). You can think of \(\beta_t\) as a 1-form on \(S_t\), and \(u_t\) as a real-valued function on \(S_t\). The contact condition then becomes more complicated, because then \(d\beta = d\beta_t + dt \wedge \frac{\partial \beta_t}{\partial t}\). Here we write \(d\beta_t\) for the 2-form on \(S_t\) which arises by taking the differential of a 1-form on \(S_t\); but \(\beta\) also has a \(t\)-dependence, and so we also obtain a derivative with respect to \(t\). Let us write \(\dot{\beta}_t\) for \(\frac{\partial \beta_t}{\partial t}\). Then the contact condition is
\[
\alpha \wedge d\alpha
= ( u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t ) \wedge dt .
\]
So in this more general case, with \(\beta\) and \(u\) depending on \(t\), the condition for \(\alpha\) to be a contact form is that \(u_t \; d\beta_t + \beta_t \wedge (du_t – \dot{\beta}_t )\) be an area form on \(S\).

So the answer to the question may not be clear. When \(\alpha\) takes the form \(\beta_t + u_t \; dt\), even if each surface \(S_t\) has a dividing set, with Liouville structures on either side, this only means that on each slice we have the first condition above, that \(u_t \; d\beta_t + \beta_t \wedge du_t\) is an area form on \(S_t\). To show that we have a contact form, we need to show the latter condition, that \(u_t \; d\beta_t + \beta_t \wedge (du_t – \dot{\beta}_t)\). The term with a \(\dot{\beta}_t\), taking a derivative in the \(t\)-direction, makes a difference.

But in any case, Giroux shows the answer is yes. This is his “realisation lemma”, which is lemma 2.4 of the 2000 paper. And the proof is not too difficult. Let’s state the result and prove it.

REALISATION LEMMA (Giroux): Let \(\beta_t\) be a family of 1-forms on \(S\), and \(v_t\) a family of functions \(S \rightarrow \mathbb{R}\), such that for all \(t\),
\[
v_t \; d\beta_t + \beta_t \wedge dv_t
\]
is an area form on \(S\). Then there exists a contact structure on \(S \times [-1,1]\) with a contact form of the form \(\beta_t + u_t \; dt\), where each \(u_t\) is a real-valued function on \(S\).

In other words, if each \(S_t\) is divided by the foliation \(\mathcal{F}_t\), then there exists a contact structure on \(S \times [-1,1]\) with \(\mathcal{F}_t\) as its movie of characteristic foliations.

PROOF:
Choose an orientation on \(S\) which agrees with \(v_t \; d\beta_t + d\beta_t \wedge dv_t\), so that we can write
\[
v_t \; d\beta_t + d\beta_t \wedge dv_t > 0.
\]
Now we need to find functions \(u_t\) such that
\[
u_t \; d\beta_t + \beta_t \wedge ( du_t – \dot{\beta}_t ) > 0.
\]
Clearly, the only difference between these two inequalities is the term \(\beta_t \wedge \dot{\beta}_t\). But the \(\beta_t\) are fixed — it’s the \(u_t\) we get to choose. And \(S \times [-1,1]\) is a compact set. So \(\beta_t \wedge \dot{\beta}_t\) only gets so large.

Similarly, by compactness, \(v_t \; d\beta_t + d \beta_t \wedge dv_t\), as a positive area form, only gets so small. If we multiply \(v_t\) by a large constant \(K\), then \(v_t \; d\beta_t + d \beta_t \wedge dv_t\) is also gets multiplied by \(K\) — and thus can be guaranteed to be arbitrarily large everywhere. Indeed, we can make it so large that it overwhelms the term \(\beta_t \wedge \dot{\beta}_t\).

And that is what we do. Let \(u_t = K v_t\) for sufficiently large \(K>0\). This is all we need to do. QED.

(This proof assumes the \(v_t\) vary smoothly in \(t\). But even if the \(v_t\) don’t vary smoothly, or even continuously, in \(t\), one can use a partition of unity to construct the desired \(u_t\).)