Symplectisation of a contact manifold

Intuition [Beginner]

The symplectisation is a recipe that lifts a contact manifold one dimension up into the symplectic world. Take an odd-dimensional contact manifold $M$. Multiply it by a copy of the real line. Put on the resulting even-dimensional product a symplectic form built from the contact form on $M$ and the new line coordinate. The output is a symplectic manifold whose geometry remembers the contact geometry of $M$ on every horizontal slice and whose vertical direction acts by a uniform rescaling.

The reason the construction matters: contact and symplectic geometry are two sides of one subject, and the symplectisation is the bridge that lets every contact question lift to a symplectic question. Floer-type theories that count Reeb orbits on $M$ run their pseudoholomorphic curves on the symplectisation, where the cylindrical symmetry tames the analysis.

A useful image: take a sphere with its contact structure and stack copies of it along a vertical axis. Each horizontal slice is a copy of the sphere; the stack is a cylinder; the symplectic form pairs the horizontal twist of the contact planes with vertical translation. The cylinder is the symplectisation.

Visual [Beginner]

A schematic of the cylinder $M\times \mathbb{R}$ as a stack of horizontal copies of a contact $3$-manifold, with the vertical $t$-axis labelled and arrows indicating the rescaling action $(p,t)\mapsto (p,t+s)$ that scales the symplectic form by $e^s$.

The key picture is the cylinder: the contact manifold sits as a level set, the $\mathbb{R}$-direction adds the missing even dimension, and the symplectic form pairs the two.

Worked example [Beginner]

Take the unit circle $S^1\subset {\mathbb{R}}^2$ with the one-form $\alpha =xdy-ydx$ pulled back from the plane. Restricted to $S^1$, this one-form is a contact form on a one-dimensional manifold (a degenerate case where the contact condition reduces to $\alpha$ never vanishing). Its symplectisation is the cylinder $S^1\times \mathbb{R}$ with coordinates $(\theta ,t)$ and the symplectic form $d(e^{t}d\theta )=e^{t}dt\wedge d\theta$.

This is the standard symplectic form on the half-plane after the change of variables $r=e^t$: $e^{t}dt\wedge d\theta$ becomes $dr\wedge d\theta$ on $\{r>0\}$, which is the area form on ${\mathbb{R}}^2$ minus the origin in polar coordinates. So the symplectisation of the circle is the punctured plane.

The same picture in higher dimensions: the symplectisation of the standard contact sphere $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ is ${\mathbb{C}}^{n+1}$ with the origin removed, and the symplectic form on the cylinder pulls back to the standard symplectic form via the radial coordinate.

What this tells us: the symplectisation of a contact manifold is a concrete cylindrical symplectic manifold, and the standard examples reproduce familiar punctured Euclidean spaces with their canonical symplectic forms.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $(M,\xi )$ be a co-orientable contact manifold of dimension $2n+1$ with a chosen contact form $\alpha$ (so $\xi =\ker \alpha$ and $\alpha \wedge (d\alpha {)}^n\ne 0$). The symplectisation of $(M,\alpha )$ is the symplectic manifold

$$(M\times \mathbb{R},\omega =d(e^t\alpha )),$$

where $t$ denotes the coordinate on $\mathbb{R}$ and $e^t\alpha$ is shorthand for the one-form on $M\times \mathbb{R}$ obtained by pulling $\alpha$ back along the projection $M\times \mathbb{R}\to M$ and multiplying by $e^t$.

Verification that $\omega$ is symplectic. Compute

$$\omega =d(e^t\alpha )=e^t(dt\wedge \alpha +d\alpha ).$$

The form is closed by construction. To check non-degeneracy, raise to the top power: on a manifold of dimension $2n+2$ the relevant power is $(n+1)$, and

$${\omega }^{n+1}=(n+1)!e^{(n+1)t}dt\wedge \alpha \wedge (d\alpha {)}^n.$$

The factor $\alpha \wedge (d\alpha {)}^n$ is the contact volume form on $M$ — non-vanishing exactly because $\alpha$ is a contact form — and $dt$ is non-vanishing on $\mathbb{R}$. Hence ${\omega }^{n+1}$ never vanishes, and $\omega$ is symplectic. The contact condition on $\alpha$ is exactly the symplectic condition on $\omega$.

Independence of the contact form. Suppose ${\alpha }^{\prime }=f\alpha$ for some smooth $f:M\to {\mathbb{R}}_{>0}$. The map

$${\Phi }_f:M\times \mathbb{R}⟶M\times \mathbb{R},(p,t)\mapsto (p,t-\log f(p)),$$

is a diffeomorphism, and a direct computation gives ${\Phi }_f^{\ast }(e^t{\alpha }^{\prime })=e^{t-\log f(p)}f(p)\alpha =e^t\alpha$, so ${\Phi }_f^{\ast }{\omega }^{\prime }=\omega$. The symplectisation depends on the contact structure $\xi$ only, not on the choice of contact form.

Liouville structure. The form $\omega =d\lambda$ is exact with primitive

$$\lambda =e^t\alpha .$$

The associated Liouville vector field $X_{\lambda }$ is the unique vector field satisfying ${\iota }_{X_{\lambda }}\omega =\lambda$. A computation in coordinates: setting $X_{\lambda }={\partial }_t$, one has ${\iota }_{{\partial }_t}(e^t(dt\wedge \alpha +d\alpha ))=e^t\alpha =\lambda$, so $X_{\lambda }={\partial }_t$. The Liouville vector field integrates to translation $(p,t)\mapsto (p,t+s)$, which scales $\lambda$ by $e^s$ and $\omega$ by $e^s$ as well — the symplectisation carries a one-parameter group of conformal symplectic dilations.

Reeb flow lifts. The Reeb vector field $R_{\alpha }$ of the contact form on $M$ extends to a vector field on $M\times \mathbb{R}$ (regarded as a section of $TM\subset T(M\times \mathbb{R})$), and a direct check gives

$${\iota }_{R_{\alpha }}\omega =e^t({\iota }_{R_{\alpha }}(dt\wedge \alpha )+{\iota }_{R_{\alpha }}d\alpha )=e^t(\alpha (R_{\alpha })dt)=e^{t}dt=d(e^t).$$

So $R_{\alpha }$ is the Hamiltonian vector field on the symplectisation for the Hamiltonian function $H=e^t$.

Key theorem with proof [Intermediate+]

Theorem (symplectisation is well-defined and $\xi$-natural). Let $(M,\xi )$ be a co-orientable contact manifold of dimension $2n+1$. For any choice of contact form $\alpha$ defining $\xi$, the pair $(M\times \mathbb{R},d(e^t\alpha ))$ is an exact symplectic manifold of dimension $2n+2$. For any two contact forms $\alpha ,{\alpha }^{\prime }$ defining the same $\xi$, the symplectisations are symplectomorphic via an explicit diffeomorphism that translates the $\mathbb{R}$-coordinate by $-\log ({\alpha }^{\prime }/\alpha )$.

Proof. Existence and exactness are immediate from the formula $\omega =d(e^t\alpha )$. The non-degeneracy computation goes:

$${\omega }^{n+1}=(e^t(dt\wedge \alpha +d\alpha ){)}^{n+1}=e^{(n+1)t}\cdot (\genfrac{}{}{0ex}{}{n+1}{1})dt\wedge \alpha \wedge (d\alpha {)}^n,$$

where the binomial expansion of the wedge picks out only the term containing the $dt$-factor exactly once and the $d\alpha$-factor exactly $n$ times (the $\alpha$-factor squares to zero, and the $dt$-factor squares to zero, so all other terms vanish). Up to the constant $(n+1)!$ which absorbs the binomial after wedge-symmetrisation, this is $e^{(n+1)t}dt\wedge \alpha \wedge (d\alpha {)}^n$. Non-vanishing of ${\omega }^{n+1}$ on $M\times \mathbb{R}$ reduces to non-vanishing of $\alpha \wedge (d\alpha {)}^n$ on $M$, which is the contact condition.

For independence from the contact form: write ${\alpha }^{\prime }=f\alpha$ with $f>0$ (since $\alpha ,{\alpha }^{\prime }$ both define the same co-orientation $\xi$, the function $f$ has constant sign, and the co-orientation choice fixes $f>0$). Define $\Phi :M\times \mathbb{R}\to M\times \mathbb{R}$ by $\Phi (p,t)=(p,t-\log f(p))$. Then

$${\Phi }^{\ast }(e^t{\alpha }^{\prime })=e^{t-\log f(p)}\cdot f(p)\alpha =e^t\alpha ,$$

since ${\Phi }^{\ast }{\alpha }^{\prime }={\alpha }^{\prime }$ on the $M$-factor and ${\Phi }^{\ast }(e^t){|}_{(p,t)}=e^{t-\log f(p)}=e^t/f(p)$. Taking exterior derivatives gives ${\Phi }^{\ast }d(e^t{\alpha }^{\prime })=d(e^t\alpha )$, and $\Phi$ is a symplectomorphism between the two symplectisations. $\square$

Bridge. The symplectisation construction here builds toward 05.10.03 (Gray's theorem), where the same path-of-contact-forms idea appears again in a global setting: a smooth path of contact structures on a compact manifold is generated by a contact isotopy, and the proof transports through the symplectisation of each form to a parametric Moser argument 05.01.05. The conformal action of the Liouville vector field ${\partial }_t$ is exactly the symplectic-Moser flow specialised to the cylindrical symmetry, and the foundational reason every cylindrical Floer theory works in $M\times \mathbb{R}$ is exactly that $\omega =d(e^t\alpha )$ identifies the $\mathbb{R}$-translation symmetry with a symplectic conformal scaling — putting these together, the symplectisation is the bridge from contact dynamics on $M$ to symplectic dynamics on $M\times \mathbb{R}$, and the bridge is what makes Symplectic Field Theory and Embedded Contact Homology possible. The construction generalises the Liouville cone over the boundary of any symplectic manifold of contact type.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not yet name contact manifolds, contact forms, or the symplectisation construction. A proposed signature:

[object Promise]

The construction depends on names that are not currently present in Mathlib (the wedge product on smooth differential forms with manifold coordinates, the contact-condition predicate as a smooth section, the product-manifold smooth structure on $M\times \mathbb{R}$). Each is a candidate Mathlib contribution.

Advanced results [Master]

The symplectisation organises the contact-symplectic adjacency into a single functor: contact manifolds map to exact symplectic manifolds with a Liouville structure, contact forms map to Liouville one-forms, contactomorphisms lift to Liouville-equivariant symplectomorphisms, and the contact dynamics of Reeb flows lift to Hamiltonian dynamics on the cylinder.

Liouville structure. The symplectisation $(M\times \mathbb{R},\omega =d\lambda )$ comes equipped with the Liouville one-form $\lambda =e^t\alpha$ and the Liouville vector field $X_{\lambda }={\partial }_t$ characterised by ${\iota }_{X_{\lambda }}\omega =\lambda$. The flow of $X_{\lambda }$ is the translation $(p,t)\mapsto (p,t+s)$, which scales $\lambda$ by $e^s$ and $\omega$ by $e^s$. The triple $(M\times \mathbb{R},\omega ,X_{\lambda })$ is the prototype Liouville manifold; every Liouville manifold with cylindrical end has the symplectisation of a contact manifold as its end model.

Contactisation, the inverse construction. Going the other direction: let $(W,\omega =d\lambda )$ be an exact symplectic manifold with Liouville vector field $X_{\lambda }$ defined by ${\iota }_{X_{\lambda }}\omega =\lambda$, and let $M\subset W$ be a hypersurface transverse to $X_{\lambda }$. Then $\alpha :=\lambda {|}_M$ is a contact form on $M$, the contact condition $\alpha \wedge (d\alpha {)}^n\ne 0$ following from non-degeneracy of $\omega$ on the transverse slice. The flow of $X_{\lambda }$ parametrises a neighbourhood of $M$ as $M\times (-ϵ,ϵ)$, and the pullback of $\lambda$ under this parametrisation is $e^t\alpha$ — so a neighbourhood of $M$ in $W$ recovers the symplectisation of $(M,\alpha )$ near level $t=0$. This is the contact-type hypersurface construction that produces every exact symplectic manifold's contact boundary.

Reeb flow as Hamiltonian flow. The Reeb vector field $R_{\alpha }$ on $M$ extends to $M\times \mathbb{R}$ by parallel transport in the $\mathbb{R}$-direction. The extension satisfies ${\iota }_{R_{\alpha }}\omega =-d(e^t)$ (with sign conventions as in Exercise 4), hence $R_{\alpha }$ is the Hamiltonian vector field of $H=-e^t$ on the symplectisation. More generally, time-dependent Hamiltonians of the form $H_t=h(t)e^t$ on the symplectisation produce Reeb-like flows that interpolate between the Reeb flows of different contact forms, and this freedom is the technical foundation of cylindrical contact homology.

Floer theory in symplectisations. Pseudoholomorphic curves $u:(\Sigma ,j)\to (M\times \mathbb{R},J)$ for an almost complex structure $J$ compatible with $\omega$ and adapted to the cylindrical structure (translation-invariant in $t$, sending ${\partial }_t$ to $R_{\alpha }$) form the central analytic object of contact Floer theories. The cylindrical Cauchy-Riemann equation reads ${\partial }_{s}u+J({\partial }_{\tau }u)=0$ on a Riemann surface with cylindrical ends, and the asymptotic limits of $u$ at the ends are closed Reeb orbits on $M$. This setup defines:

• Symplectic Field Theory (SFT, Eliashberg-Givental-Hofer 2000): a graded algebra of formal power series in Reeb-orbit generators, with differential counting genus-zero pseudoholomorphic curves with multiple punctures in symplectisations and symplectic cobordisms between contact manifolds.
• Cylindrical Contact Homology: the simplest piece of SFT, generated by good Reeb orbits with differential counting cylinders between them.
• Embedded Contact Homology (ECH, Hutchings 2002): a refinement in dimension three counting embedded curves with strict bounds, connected to Seiberg-Witten Floer theory by Taubes.

In each case, the symplectisation is the setting where the analysis is run; the symplectic geometry of $M\times \mathbb{R}$ is the technical replacement for the contact dynamics on $M$.

Liouville cobordisms. Symplectic manifolds whose ends are symplectisations of contact manifolds — Liouville cobordisms — are the morphisms in the Liouville category. The composition of two such cobordisms is gluing along a common contact end, performed by truncating the symplectisation cylinders and identifying. The resulting category is the natural setting for SFT functoriality: the SFT of a contact manifold $M$ is the "endomorphism algebra" of the symplectisation $M\times \mathbb{R}$, and Liouville cobordisms induce module homomorphisms between SFTs.

Failure modes.

• Non-co-orientable contact structures. When $\xi$ is not co-orientable, no global contact form $\alpha$ exists, and the symplectisation construction does not directly apply. The double cover that co-orients $\xi$ admits a symplectisation, and the deck transformation acts by $(p,t)\mapsto (-p,-t)$ on the symplectisation; the quotient by this $\mathbb{Z}/2$-action is a singular symplectic space.
• Confusion with contactisation. Contactisation goes the other way: an exact symplectic manifold $(W,\omega =d\lambda )$ produces a contact manifold $(W\times \mathbb{R},dt+{\pi }^{\ast }\lambda )$, where $t$ is the new coordinate. This is not the inverse of symplectisation in the strict sense — symplectisation produces a manifold of two more dimensions than the contact base, while contactisation produces a manifold of one more dimension than the symplectic base. The two functors are adjoint at the level of categories of forms, not at the level of manifolds.
• Sign conventions. Different texts use $d(e^t\alpha )$ versus $d(e^{-t}\alpha )$; the latter swaps the role of the two ends of the cylinder. Geiges and Cannas use $d(e^t\alpha )$; some SFT papers use $-t$ to make the Reeb-orbit count run from $t=+\infty$ down to $t=-\infty$. The choice is a global sign on the Liouville vector field.

Synthesis. The symplectisation is exactly the bridge between contact and symplectic geometry: a contact manifold $M$ generates a symplectic manifold $M\times \mathbb{R}$, and every contact-geometric question on $M$ becomes a $\mathbb{R}$-equivariant symplectic question on the cylinder. This is precisely the same functorial relationship that recurs throughout symplectic topology — the Liouville cone over a contact-type boundary, the cylindrical end of an exact symplectic filling, the affine cone in algebraic geometry where a projective variety with a polarisation produces an affine variety with a ${\mathbb{C}}^{\ast }$-action. Putting these together, the foundational reason cylindrical Floer theory works is exactly that the symplectisation identifies contact forms with Liouville one-forms — and the Reeb dynamics on $M$ with the Hamiltonian dynamics of $e^t$ on $M\times \mathbb{R}$. Read in the opposite direction, the contactisation of an exact symplectic manifold specialises the symplectisation to the case of a single contact-type hypersurface, and in this sense the symplectisation generalises the boundary correspondence between symplectic manifolds with contact-type boundary and their boundary contact manifolds. The bridge is the identification $\omega =d(e^t\alpha )$, and this single formula identifies contact geometry with cylindrical exact symplectic geometry.

Full proof set [Master]

Lemma (non-degeneracy of $\omega$). On the symplectisation $M\times \mathbb{R}$ with $\omega =d(e^t\alpha )$, the form ${\omega }^{n+1}$ is a nowhere-vanishing top-degree form, equal to $(n+1)!e^{(n+1)t}dt\wedge \alpha \wedge (d\alpha {)}^n$ up to constants.

Proof. Expand $\omega =e^t(dt\wedge \alpha +d\alpha )$. The $(n+1)$-fold wedge ${\omega }^{n+1}$ factors through the binomial expansion of $(dt\wedge \alpha +d\alpha {)}^{n+1}$. Two facts kill most terms: $(dt\wedge \alpha {)}^2=-dt\wedge dt\wedge \alpha \wedge \alpha =0$ since both $dt$ and $\alpha$ square to zero in the wedge, and any term with two or more $dt$-factors vanishes for the same reason. So the only surviving term has exactly one $(dt\wedge \alpha )$-factor and $n$ copies of $d\alpha$. The binomial coefficient $\left(\genfrac{}{}{0ex}{}{n+1}{1}\right)=n+1$ counts these terms; the wedge gives $(n+1)\cdot dt\wedge \alpha \wedge (d\alpha {)}^n$. Pulling out the scalar $e^t$ raised to the $(n+1)$ gives ${\omega }^{n+1}=(n+1)e^{(n+1)t}dt\wedge \alpha \wedge (d\alpha {)}^n$ up to overall sign and the $(n!)$-factor that absorbs the wedge-symmetrisation. The contact condition $\alpha \wedge (d\alpha {)}^n\ne 0$ on $M$ then makes ${\omega }^{n+1}$ nowhere zero on $M\times \mathbb{R}$. $\square$

Lemma (the conformal-rescaling diffeomorphism). For ${\alpha }^{\prime }=f\alpha$ with $f>0$, the diffeomorphism ${\Phi }_f(p,t)=(p,t-\log f(p))$ satisfies $\Phi_f^ \lambda' = \lambda$where$\lambda' = e^t \alpha'$and$\lambda = e^t \alpha$.*

Proof. Let $g=-\log f:M\to \mathbb{R}$. Then ${\Phi }_f$ acts on the $\mathbb{R}$-coordinate by $t\mapsto t+g(p)$ and as the identity on the $M$-coordinate. Pulling back $e^t$: ${\Phi }_f^{\ast }(e^t){|}_{(p,t)}=e^{t+g(p)}=e^t/f(p)$ (substituting $g=-\log f$). Pulling back ${\alpha }^{\prime }$: ${\Phi }_f^{\ast }{\alpha }^{\prime }={\alpha }^{\prime }$ since ${\alpha }^{\prime }$ is pulled up from the $M$-factor and ${\Phi }_f$ acts as the identity there. Combining: ${\Phi }_f^{\ast }(e^t{\alpha }^{\prime })=(e^t/f)\cdot f\alpha =e^t\alpha =\lambda$. Taking $d$ on both sides gives ${\Phi }_f^{\ast }{\omega }^{\prime }=\omega$. $\square$

Theorem (Liouville-flow trivialisation). Let $X_{\lambda }={\partial }_t$ be the Liouville vector field on the symplectisation $(M\times \mathbb{R},\omega =d\lambda )$. The flow ${\varphi }_s$ of $X_{\lambda }$ is ${\varphi }_s(p,t)=(p,t+s)$, and $\phi_s^ \lambda = e^s \lambda$,$\phi_s^* \omega = e^s \omega$.*

Proof. The vector field ${\partial }_t$ has the obvious flow ${\varphi }_s(p,t)=(p,t+s)$. Pull back $\lambda =e^t\alpha$: ${\varphi }_s^{\ast }(e^t\alpha )=e^{t+s}\alpha =e^s\cdot e^t\alpha =e^s\lambda$. Differentiating gives ${\varphi }_s^{\ast }\omega =d(e^s\lambda )=e^{s}d\lambda =e^s\omega$. The infinitesimal version is ${\mathcal{L}}_{X_{\lambda }}\lambda =\lambda$ and ${\mathcal{L}}_{X_{\lambda }}\omega =\omega$, which also follow from Cartan's formula and the Liouville defining equation ${\iota }_{X_{\lambda }}\omega =\lambda$ together with $d\omega =0$. $\square$

Theorem (contact-type hypersurface). Let $(W,\omega =d\lambda )$ be an exact symplectic manifold of dimension $2n+2$ with Liouville vector field $X_{\lambda }$, and let $M\subset W$ be a closed hypersurface transverse to $X_{\lambda }$. Then $\alpha = \iota^ \lambda$isacontactformon$M$,where$\iota : M \hookrightarrow W$istheinclusion.Theflowof$X_\lambda$trivialisesatubularneighbourhoodof$M$in$W$asthelevel$|t| < \epsilon$sliceofthesymplectisationof$(M, \alpha)$.*

Proof. For the contact condition, compute $\alpha \wedge (d\alpha {)}^n={\iota }^{\ast }(\lambda \wedge (d\lambda {)}^n)$. The form $\lambda \wedge (d\lambda {)}^n$ on $W$ pulls back to ${\iota }_{X_{\lambda }}{\omega }^{n+1}$ via the Liouville identity, and transversality of $X_{\lambda }$ to $M$ together with non-degeneracy of ${\omega }^{n+1}$ on $W$ gives ${\iota }^{\ast }({\iota }_{X_{\lambda }}{\omega }^{n+1})\ne 0$ on $M$. Hence $\alpha \wedge (d\alpha {)}^n\ne 0$, and $\alpha$ is contact.

For the tubular neighbourhood: the flow ${\varphi }_t$ of $X_{\lambda }$ on $W$ defines a smooth map $\Psi :M\times (-ϵ,ϵ)\to W$ by $\Psi (p,t)={\varphi }_t(p)$. Transversality makes $\Psi$ a diffeomorphism onto a tubular neighbourhood. Pull back $\lambda$: by the Liouville-flow trivialisation, ${\varphi }_t^{\ast }\lambda =e^t\lambda {|}_{{\varphi }_t^{-1}}$, so on $M\times (-ϵ,ϵ)$ the pullback ${\Psi }^{\ast }\lambda$ at the point $(p,t)$ is $e^t\cdot \alpha$, where $\alpha =\lambda {|}_M$. Hence ${\Psi }^{\ast }\omega =d(e^t\alpha )$, which is the symplectisation form. $\square$

Connections [Master]

• Contact manifold 05.10.01. The symplectisation is the canonical lift of a contact manifold to a symplectic manifold one dimension up. Every contact-geometric question on $(M,\xi )$ has a symplectic-geometric translation on $(M\times \mathbb{R},d(e^t\alpha ))$, and this is the bridge between the two parallel subjects.

• Symplectic manifold 05.01.02. The symplectisation is an exact symplectic manifold with a distinguished Liouville structure; conversely, the contact-type hypersurfaces of exact symplectic manifolds recover the contact base by restriction. The two constructions are adjoint at the categorical level — exact symplectic manifolds with contact-type boundary versus their contact boundaries.

• Moser's trick 05.01.05. Gray's theorem on contact stability proceeds by lifting the contact-form path to a path of Liouville one-forms on the symplectisation and applying the Moser-trick template there. The contact-Moser equation is exactly the symplectic-Moser equation reduced by the conformal action of the Liouville vector field on the symplectisation.

• Gray's theorem 05.10.03. A smooth path of contact structures on a compact $M$ is generated by a contact isotopy. The proof works in the symplectisation: the path of contact forms produces a path of Liouville one-forms on $M\times \mathbb{R}$, and the parametric Moser argument on the cylinder descends to the contact isotopy on $M$.

• Floer homology 05.08.02. Cylindrical contact homology, Symplectic Field Theory, and Embedded Contact Homology all run pseudoholomorphic curves on the symplectisation of a contact manifold. The translation-invariant almost complex structure adapted to the Reeb dynamics is the analytic foundation of every Reeb-orbit-counting Floer theory.

• Hamiltonian vector field 05.02.01. The Reeb vector field on $M$ lifts to the Hamiltonian vector field of $H=e^t$ on the symplectisation, identifying contact dynamics with a special class of Hamiltonian dynamics on the cylinder. This identification is the technical content of the assertion "contact dynamics is symplectic dynamics with a conformal symmetry."

Historical & philosophical context [Master]

The symplectisation construction is implicit in mid-twentieth-century contact-geometry literature. Georges Reeb's 1952 Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques ^{[Reeb 1952]} introduced the contact-form-and-Reeb-flow framework, and the cone construction $M\times {\mathbb{R}}_{>0}$ over a contact manifold was a routine tool in the differential-geometric tradition that followed.

The modern framing — symplectisation as a functor, Liouville structure as the organising data, cylindrical Floer theory as the application — emerged in the 1990s. Yakov Eliashberg's contact-topology programme, building on Bennequin (1983) and Eliashberg's own classification work ^{[Eliashberg 1989]}, placed the symplectisation at the centre of the contact-symplectic adjacency. Helmut Hofer's introduction of pseudoholomorphic-curve methods to contact geometry (Hofer 1993, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture) made the symplectisation the analytic setting for the Weinstein conjecture, proving the conjecture for overtwisted contact 3-manifolds and for $S^3$ with the standard form.

Eliashberg-Givental-Hofer's Introduction to Symplectic Field Theory (2000, GAFA special volume) ^{[Eliashberg-Givental-Hofer 2000]} formalised the use of symplectisations as the universal local model for Floer-theoretic invariants of contact manifolds and proposed the SFT framework with its full algebraic structure. Michael Hutchings's 2002 paper introducing Embedded Contact Homology ^{[Hutchings 2002]} specialised the SFT framework to dimension three and to embedded curves, producing an invariant later identified with Seiberg-Witten Floer cohomology by Taubes (2007-2010).

Cieliebak-Eliashberg's From Stein to Weinstein and Back (2012) ^{[Cieliebak-Eliashberg 2012]} reorganised the symplectic geometry of affine complex manifolds around the Liouville-structure framework, treating symplectisations as one piece of the larger Liouville-cobordism category. Hans Geiges's An Introduction to Contact Topology (Cambridge, 2008) ^[Geiges] is the standard pedagogical treatment from the contact-geometry side, and Cannas da Silva's Lectures on Symplectic Geometry §10 ^{[Cannas da Silva]} gives the construction within the symplectic-geometry pedagogical arc.

Bibliography [Master]

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