Symplectic Geometry

Motivation

A contact structure on a manifold of odd dimension $2k+1$ is a field of hyperplanes $\xi=\ker\alpha$ that is as far from integrable as possible. Contact structures are the odd-dimensional counterpart of symplectic forms. They arise on every star-shaped energy level of a Hamiltonian system, on unit cotangent bundles (geodesic flows), and on links of singularities. They are the setting of Reeb dynamics and of the Weinstein conjecture.

Gray's stability theorem says that on a closed manifold contact structures have no local moduli. If $\xi_t$, $t\in[0,1]$, is a smooth family of contact structures, there is an isotopy $\psi_t$ with $T\psi_t(\xi_0)=\xi_t$. A contact structure can therefore be deformed only within its isotopy class, and every classification of contact structures (tight versus overtwisted, Eliashberg's classification on $S^3$) is a classification up to isotopy because of it.

Timeline.

• 1959. Gray proves stability with deformation theory in the style of Kodaira–Spencer (Ann. of Math. 69).
• 1965. Moser proves the analogous stability for volume forms by integrating a time-dependent vector field, now called the Moser trick (Trans. AMS 120).
• Later. The Moser trick becomes the standard proof of Gray's theorem. Geiges' survey gives a short complete proof (arXiv:math/0307242, Theorem 2.20), which is the source of this mission. Its remarks record the limits of the statement: contact forms are not stable (Remark 2.21(1)), and on the open manifold $S^1\times\mathbb{R}^2$ stability fails (Remark 2.21(2), after Eliashberg).

Setting

The closed manifold is a compact submanifold of Euclidean space. Let $F:\mathbb{R}^n\to\mathbb{R}^c$ be smooth, let $M=F^{-1}(0)$ be compact, and assume $DF(y)$ is surjective for every $y\in M$. Then $M$ is a closed smooth manifold of dimension $n-c$ with tangent spaces

$$T_yM=\ker DF(y).$$

A one-form is a smooth map $\alpha:\mathbb{R}^n\to(\mathbb{R}^n)^*$, restricted to $TM$. Its exterior derivative is

$$d\alpha_y(u,v)=D\alpha_y(u)(v)-D\alpha_y(v)(u).$$

• Contact form. $\alpha$ is a contact form on $M$ if at every $y\in M$ the covector $\alpha_y$ is nonzero on $T_yM$ and $d\alpha_y$ is non-degenerate on the hyperplane

$$\xi_y=T_yM\cap\ker\alpha_y.$$

This is the condition $\alpha\wedge(d\alpha)^k\neq0$ in the form of Geiges' Remark 2.3. It forces $\dim M$ to be odd. The contact structure is $\xi=\ker\alpha$. Contact structures are cooriented throughout, as in Geiges' standing assumption (§2).

• Smooth family. A family $\alpha_t$ is smooth if $(t,y)\mapsto\alpha_t(y)$ is smooth. It is a family of contact forms if each $\alpha_t$, $t\in[0,1]$, is a contact form on $M$.
• Isotopy. An isotopy of $M$ is a smooth map $(t,y)\mapsto\psi_t(y)$ with $\psi_0=\mathrm{id}$ on $M$, such that each $\psi_t$, $t\in[0,1]$, maps $M$ bijectively onto $M$ with injective differential on $TM$, i.e. is a diffeomorphism of $M$.
• Pull-back. $(\psi_t^*\alpha)_y(v)=\alpha_{\psi_t(y)}(D\psi_t(y)\,v)$.

Formalization targets

Goal: Gray stability (Theorem 2.20)

For every smooth family of contact forms $\alpha_t$, $t\in[0,1]$, on $M$ there is an isotopy $\psi_t$ of $M$ with

$$T\psi_t(\xi_0)=\xi_t\qquad(t\in[0,1]),\qquad\xi_t=\ker\alpha_t,$$

stated pointwise: for $v\in T_yM$, $\alpha_0(v)=0$ iff $\alpha_t(D\psi_t(y)v)=0$.

The goal concerns the contact structures. It fixes no normalization of the forms and no dimension.

Stronger: conformal form

The same isotopy can be chosen with smooth functions $\lambda_t>0$ such that

$$\psi_t^*\alpha_t=\lambda_t\,\alpha_0\quad\text{on }TM.$$

Stronger: stationary points (Remark 2.21(3))

Moreover, every point $p\in M$ at which $\dot\alpha_t$ vanishes on $T_pM$ for all $t$ stays fixed: $\psi_t(p)=p$.

Significance

The result. Gray stability is the basic rigidity statement of contact topology.

• It reduces the classification of contact structures to isotopy classes, so invariants of a contact manifold are constant along deformations.
• It is the first step of many local normal forms: Darboux's theorem and the neighbourhood theorems for Legendrian and transverse submanifolds are proved by applying it, or its proof, near a submanifold (Geiges §2.4–2.5).
• In Hamiltonian dynamics it identifies the contact structures of a continuous family of star-shaped energy levels. Topological invariants of transverse periodic orbits, such as the self-linking number, are then constant along the family.

Formalizing it. The theorem is classical and has a short proof on paper. Mathlib, at the revision used here, has no differential forms on manifolds, no Lie derivative, no global flows of time-dependent vector fields on compact submanifolds, and no contact structures. The mission builds these concretely on submanifolds of $\mathbb{R}^n$:

• one-forms, their exterior derivative and pull-back;
• the derivative of a pulled-back family along a flow (Lemma 2.19);
• the pointwise linear algebra of a contact form: the Reeb vector, and the unique solution of the Moser equation;
• global flows of smooth time-dependent vector fields tangent to a compact submanifold.

All of these are reusable for Moser's theorem on volume and symplectic forms and for the Darboux and neighbourhood theorems.

Difficulty

The proof is soft, but two steps are not formal.

1. Solving for the vector field. Writing $\psi_t$ as the flow of $X_t$, the equation $\psi_t^*\alpha_t=\lambda_t\alpha_0$ becomes

$$\dot\alpha_t+i_{X_t}d\alpha_t=\mu_t\alpha_t,\qquad X_t\in\xi_t.$$

It has a unique solution at each point, but only because $d\alpha_t$ is non-degenerate on $\xi_t$ and the Reeb vector spans the kernel of $d\alpha_t|_{TM}$. The solution must also depend smoothly on $(t,y)$ and be tangent to $M$; the ambient form $\alpha_t$ is in general not contact off $M$. 2. Integrating it. The isotopy is the flow of $X_t$, which must exist for all $t\in[0,1]$ and stay on $M$. Compactness of $M$ enters exactly here. On open manifolds the statement is false (Remark 2.21(2)).

The tempting shortcut of asking for $\psi_t^*\alpha_t=\alpha_0$ does not work. Contact forms themselves are not stable, as the Hopf family on $S^3$ shows (Remark 2.21(1)), and the conformal factor $\lambda_t$ cannot be dropped.

Formalization scope

• Representation.
  • $\mathbb{R}^n$ is Fin n → ℝ; $M$ is a compact regular level set $F^{-1}(0)$ of a smooth $F:\mathbb{R}^n\to\mathbb{R}^c$.
  • One-forms are maps $\mathbb{R}^n\to(\mathbb{R}^n\to_L\mathbb{R})$, smooth families are jointly smooth in $(t,y)$ on $\mathbb{R}\times\mathbb{R}^n$, and $d\alpha$ is the antisymmetrized derivative.
  • An isotopy is a jointly smooth $\psi:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$ that restricts, for $t\in[0,1]$, to diffeomorphisms of $M$.
• Committed conventions.
  • Contact structures are cooriented, i.e. given by global contact forms.
  • The contact condition is the non-degeneracy of $d\alpha$ on $\xi$ (Remark 2.3), not a wedge power.
  • Only the restrictions to $TM$ and the values for $t\in[0,1]$ matter.
• Scope relative to the source. Every closed manifold embeds in some $\mathbb{R}^n$, but not every closed manifold is a regular level set, since that requires a trivial normal bundle. The targets cover regular level sets, including all spheres and all star-shaped energy levels. The abstract version on any closed manifold is outside the targets until Mathlib has differential forms on manifolds.
• Ruling out vacuous encodings. The isotopy must start at the identity on $M$, map $M$ onto $M$ for every $t\in[0,1]$, and be a diffeomorphism there. Dropping any of these makes the goal trivial; for instance, the constant isotopy satisfies the goal whenever all $\xi_t$ agree. The Hopf family milestone checks that the contact condition is satisfiable and that the conformal factor is necessary.
• Contributions welcome. The linear-algebra milestones (Reeb vector, Moser equation), Lemma 2.19, global flows on compact level sets, and the Hopf-family example. Moser's theorem for volume forms would be a natural sibling result built on the same infrastructure.

Selected references

• J. W. Gray, Some global properties of contact structures, Ann. of Math. 69 (1959), 421–450. https://doi.org/10.2307/1970192
• H. Geiges, Contact geometry, in Handbook of Differential Geometry, Vol. II, Elsevier (2006), 315–382; §2.2, Lemma 2.19, Theorem 2.20, Remark 2.21. https://arxiv.org/abs/math/0307242
• H. Geiges, An Introduction to Contact Topology, Cambridge Stud. Adv. Math. 109, Cambridge Univ. Press (2008), §2.2. https://doi.org/10.1017/CBO9780511611438
• J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. https://doi.org/10.1090/S0002-9947-1965-0182927-5
• Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier 42 (1992), 165–192. https://doi.org/10.5802/aif.1288