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!