Contact Geometry

Introduction

A contact structure on an odd-dimensional manifold is a hyperplane field that is as far as possible from being tangent to a foliation: it is maximally non-integrable, so that at every point the field winds in the most efficient way and no surface of positive dimension can be tangent to it beyond the maximal allowed dimension. Equivalently, the field is the kernel of a $1$-form $\alpha$ satisfying the nondegeneracy condition $\alpha\wedge(d\alpha)^n \neq 0$ on a manifold of dimension $2n+1$, and the two descriptions are related by the Frobenius theorem. Contact geometry is the odd-dimensional companion of symplectic geometry: the contact form $\alpha$ plays the role of a symplectic potential, $d\alpha$ is nondegenerate on the contact planes, and a contact manifold is the boundary at infinity of a symplectic manifold.

This article develops the geometry of a contact manifold. It defines contact forms and contact structures, shows that a contact structure is the hyperplane field rather than the form, and establishes the elementary consequences of the definition — the dimension, the maximal non-integrability, the orientation. It treats the standard examples, the Darboux theorem for contact forms and the stability theorem of Gray, the Reeb vector field, Legendrian submanifolds, and the symplectisation that turns a contact manifold into the boundary of a symplectic one. The dynamical content of a Reeb flow belongs to Part III, where the differential equation it satisfies can be discussed; here the Reeb field is used only as a distinguished transverse vector field.

The boundaries of the article follow those of Symplectic Geometry. The symplectic structure on the cotangent bundle, the Darboux theorem for symplectic forms, the Poisson bracket and the group of symplectomorphisms are the subject of Symplectic Geometry, and are assumed; contact geometry is developed here as the odd-dimensional analogue, and the relation between the two is made precise by the symplectisation. The linear algebra of an alternating form on a vector space is that of Symplectic Forms and Poisson Brackets. The contact topology — the classification questions, the tight-overtwisted dichotomy, the Legendrian knots and the invariants of contact manifolds — is not covered here, and the Legendrian knots themselves belong to it. The differential geometry of the geodesic flow on the unit tangent bundle uses the Riemannian metric and the Levi-Civita connection of the companion article Riemannian Geometry, and the complex-geometric examples use the holomorphic structures.

Manifolds, charts and the tangent bundle are those of Smooth Manifolds and Differential Geometry and Differential Topology, being written in parallel; differential forms, the exterior derivative and the Frobenius theorem in its form-theoretic statement are those of Differential Forms and Stokes' Theorem; bundles and the tautological form of a cotangent bundle are those of Fibre Bundles, Connections and Curvature. The base field is $\mathbb{R}$ and no physics is invoked.

Contact Manifolds

Definition. A contact form on a smooth manifold $M$ of dimension $2n+1$ is a $1$-form $\alpha \in \Omega^1(M)$ such that

$$ \alpha \wedge (d\alpha)^{\wedge n} \neq 0 $$

at every point of $M$. A contact structure is the hyperplane field $\xi = \ker\alpha$ determined by a contact form, and the pair $(M, \xi)$ is a contact manifold. The field $\xi$ is also called the field of contact planes, and $d\alpha$ is the contact form's differential, which restricts to a nondegenerate form on each $\xi_x$.

The name contact records the opposite of tangency: the planes are not tangent to any surface of dimension greater than $n$, because the field is maximally non-integrable. The precise statement is the following.

Proposition. Let $\alpha$ be a $1$-form on a manifold of odd dimension $2n+1$ with $\alpha \neq 0$ everywhere. Then the hyperplane field $\xi = \ker\alpha$ is integrable if and only if $\alpha\wedge d\alpha = 0$; when this holds, $\xi$ is tangent to a foliation of codimension one. If instead $\alpha\wedge(d\alpha)^{\wedge n} \neq 0$, then $\xi$ is non-integrable, and the maximum dimension of a submanifold tangent to $\xi$ is $n$.

Proof. Suppose first that $\xi$ is integrable. By the Frobenius theorem the ideal generated by $\alpha$ is differential, so $d\alpha = \alpha\wedge\beta$ for some $1$-form $\beta$, whence $\alpha\wedge d\alpha = \alpha\wedge\alpha\wedge\beta = 0$. Conversely, if $\alpha\wedge d\alpha = 0$, then $d\alpha$ vanishes on $\xi\wedge\xi$: the identity for a $1$-form $\alpha$ and a $2$-form $\beta$, $$ (\alpha\wedge\beta)(u,v,w) = \alpha(u)\,\beta(v,w) - \alpha(v)\,\beta(u,w) + \alpha(w)\,\beta(u,v), $$ applied with $\beta = d\alpha$ and with $u, v \in \xi$, gives $\alpha(w)\,d\alpha(u,v) = 0$ for every $w$, and choosing $w$ with $\alpha(w)\neq0$ gives $d\alpha(u,v)=0$; so $\xi$ is integrable by Frobenius. Finally, if $\alpha\wedge(d\alpha)^{\wedge n} \neq 0$ then $d\alpha$ is nondegenerate on $\xi_x$: a vector $u \in \xi_x$ with $d\alpha(u, v) = 0$ for all $v \in \xi_x$ could be used to contract the volume form to zero, contradicting the nonvanishing. Hence the maximal isotropic subspaces of the symplectic space $(\xi_x, d\alpha_x)$ have dimension $n$, and a submanifold tangent to $\xi$ has tangent spaces isotropic for $d\alpha$, so its dimension is at most $n$.

Proposition. Let $\alpha$ be a contact form and let $f \in C^\infty(M)$ be nowhere vanishing. Then $f\alpha$ is a contact form, and

$$ (f\alpha)\wedge\bigl(d(f\alpha)\bigr)^{\wedge n} = f^{\,n+1}\, \alpha\wedge(d\alpha)^{\wedge n}. $$

Consequently the contact structure $\xi = \ker\alpha$ depends only on the defining form up to multiplication by a nowhere vanishing function, while the form itself does not.

Proof. One has $d(f\alpha) = df\wedge\alpha + f\,d\alpha$, so

$$ (d(f\alpha))^{\wedge n} = f^{\,n}(d\alpha)^{\wedge n} + n\,f^{\,n-1}\,(d\alpha)^{\wedge(n-1)}\wedge df\wedge\alpha, $$

all other terms containing a repeated factor $\alpha\wedge\alpha$ or $df\wedge df$, and both vanish. Multiplying by $f\alpha$ kills the second term, since it contains $\alpha\wedge\alpha$, and leaves $f^{\,n+1}\alpha\wedge(d\alpha)^{\wedge n}$.

Remark. The contact structure is therefore a geometric object, the hyperplane field, and the contact form is an auxiliary choice; a contact form is a coorientation of the field, and a contact structure is coorientable when a global defining form exists. A cooriented contact manifold is oriented by the volume form $\alpha\wedge(d\alpha)^{\wedge n}$, so the contact condition is a condition of maximal non-degeneracy in both the odd direction and the contact planes. The factor $f^{\,n+1}$ in the transformation law is the contact analogue of the conformal ambiguity of a symplectic form on a fixed hyperplane field.

Definition. A contactomorphism between contact manifolds $(M, \xi)$ and $(N, \eta)$ is a diffeomorphism $F : M \to N$ with $dF(\xi) = \eta$; for cooriented structures defined by forms, the condition is $F^*\alpha_N = f\,\alpha_M$ for some nowhere vanishing function $f$. A contact vector field is a vector field $X$ whose flow consists of contactomorphisms; equivalently, $\mathcal{L}_X\alpha = \lambda\alpha$ for some function $\lambda$.

Examples

Example (the standard contact structure on $\mathbb{R}^{2n+1}$). On $\mathbb{R}^{2n+1}$ with coordinates $x_1, \ldots, x_n, y_1, \ldots, y_n, z$, put

$$ \alpha_0 = dz + \sum_{i=1}^{n} x_i\, dy_i, \qquad d\alpha_0 = \sum_{i=1}^{n} dx_i \wedge dy_i . $$

Then $\alpha_0\wedge(d\alpha_0)^{\wedge n} = dz\wedge dx_1\wedge dy_1\wedge\cdots\wedge dx_n\wedge dy_n$, a nonvanishing volume form, so $\alpha_0$ is a contact form. The contact planes are spanned by $\partial_{x_1}, \ldots, \partial_{x_n}$ together with the vector fields $\partial_{y_i} - x_i\,\partial_z$, since $\alpha_0(\partial_{x_i}) = 0$ and $\alpha_0(\partial_{y_i} - x_i\partial_z) = x_i - x_i = 0$. In the case $n = 1$ the form is $\alpha_0 = dz + x\,dy$, and the contact planes in $\mathbb{R}^3$ turn by a full turn as one traverses a circle in the $xy$-plane, which is the geometric picture of maximal non-integrability.

Example (the standard contact structure on the sphere). Let $S^{2n+1} \subseteq \mathbb{C}^{n+1}$ be the unit sphere with the ambient coordinates $x_j, y_j$ and the radial $1$-form $\alpha = \sum_{j=1}^{n+1}(x_j\,dy_j - y_j\,dx_j)$. Its restriction to $S^{2n+1}$ is a contact form: writing $r^2 = \sum_j(x_j^2+y_j^2)$ one has $d\alpha = 2\sum_j dx_j\wedge dy_j$, and a computation in a unitary frame shows that $\alpha\wedge(d\alpha)^{\wedge n}$ is a nonvanishing multiple of the spherical volume form. The contact hyperplanes are the complex tangent spaces $TS^{2n+1}\cap i\,TS^{2n+1}$. Its Reeb vector field is the generator of the Hopf action of $U(1)$, whose orbits are the fibres of the Hopf fibration $S^{2n+1}\to\mathbb{CP}^n$; each fibre is everywhere transverse to the contact planes, so the Hopf fibres are the standard transverse circles of the structure, and not Legendrian. This is the model contact manifold, and it is the boundary of the unit ball in $\mathbb{C}^{n+1}$.

Example (the spherical cotangent bundle). Let $Q$ be a Riemannian manifold and let $ST^*Q \subseteq T^*Q$ be its spherical cotangent bundle, consisting of the covectors of unit length. The restriction of the tautological $1$-form $\theta$ of Symplectic Geometry to $ST^*Q$ is a contact form, and the resulting contact structure is the geodesic contact structure. Its Reeb vector field is the geodesic flow of the metric: the Reeb flow projects to the flow that carries a unit covector along the geodesic it determines. The Riemannian structure and the geodesic equation are those of Riemannian Geometry, being written in parallel.

Example (the $1$-jet space). Let $N$ be a smooth $n$-manifold and let $J^1(N) = T^*N\times\mathbb{R}$ with coordinates $(q, p, z)$ and the form

$$ \alpha = dz - \sum_{i=1}^{n} p_i\, dq^i . $$

Then $d\alpha = \sum_i dq^i\wedge dp_i$ is the pullback of the standard symplectic form, and $\alpha\wedge(d\alpha)^{\wedge n} = dz\wedge dq^1\wedge dp_1\wedge\cdots$ is nonvanishing; so $J^1(N)$ is contact. The $1$-jet of a smooth function $f : N \to \mathbb{R}$ is the section $j^1f(q) = (q, df_q, f(q))$, and it is Legendrian: $\alpha$ restricts to $df - df = 0$ on it. More generally the graph of a closed $1$-form $\beta$ and a function $g$ with $dg = \beta$ is Legendrian, and every Legendrian submanifold of $J^1(N)$ that is transverse to the $z$-direction is locally of this form. This is the contact analogue of the Lagrangian sections of a cotangent bundle.

Example. Every oriented $3$-manifold admits a contact structure, and in fact the standard construction of a contact structure on a $3$-manifold is by an open book decomposition; in higher dimensions every oriented odd-dimensional manifold admits an almost contact structure, and the existence and classification of the overtwisted contact structures there is a theorem of Borman, Eliashberg and Murphy. The classification of contact structures up to contactomorphism, and the distinction between tight and overtwisted structures in dimension three, belong to Symplectic and Contact Topology.

The Darboux Theorem and Stability

Theorem (Darboux for contact forms). Let $\alpha$ be a contact form on a manifold of dimension $2n+1$. About every point there is a chart $(U, x_1, \ldots, x_n, y_1, \ldots, y_n, z)$ centred at the point in which

$$ \alpha = dz + \sum_{i=1}^{n} x_i\, dy_i . $$

Consequently every contact manifold of dimension $2n+1$ is locally contactomorphic to $(\mathbb{R}^{2n+1}, \ker\alpha_0)$.

Proof sketch. Choose a chart in which $\alpha$ agrees with $\alpha_0$ at the origin, and deform $\alpha_1 = \alpha$ to $\alpha_0$ through the linear family $\alpha_t = (1-t)\alpha_0 + t\alpha_1$; each $\alpha_t$ is a contact form near the origin, because the contact condition is open and holds at the origin. One seeks a time-dependent vector field $X_t$ and a function $\lambda_t$ with

$$ \mathcal{L}_{X_t}\alpha_t + \frac{d\alpha_t}{dt} = \lambda_t\,\alpha_t, $$

which is the contact analogue of the Moser equation: the form is only determined up to a conformal factor, so the right-hand side is a multiple of $\alpha_t$ rather than zero. Writing the equation, by the identity $\mathcal{L}_{X_t}\alpha_t = \iota_{X_t}d\alpha_t + d\bigl(\alpha_t(X_t)\bigr)$, as

$$ \iota_{X_t}d\alpha_t + \dot\alpha_t + d\bigl(\alpha_t(X_t)\bigr) = \mu_t\,\alpha_t , $$

one solves it by looking for $X_t$ with $\alpha_t(X_t) = 0$, that is with $X_t$ tangent to the contact planes. Then the left-hand side has vanishing contraction with the Reeb field of $\alpha_t$, and the contact-plane component of the equation reads $\iota_{X_t}d\alpha_t = -\dot\alpha_t$ on $\ker\alpha_t$, which determines $X_t$ uniquely because $d\alpha_t$ is nondegenerate on $\ker\alpha_t$. With $X_t$ so chosen the difference $\iota_{X_t}d\alpha_t + \dot\alpha_t$ vanishes on the contact planes, hence is a multiple $\mu_t\alpha_t$ of the contact form, since the contact planes are the kernel of $\alpha_t$. The flow of $X_t$ at time $1$ is then the required contactomorphism.

Theorem (Gray). Let $\alpha_t$, $t \in [0,1]$, be a smooth family of contact forms on a closed manifold. Then there is an isotopy $\varphi_t$ with $\varphi_t^*\alpha_t = f_t\alpha_0$ for nowhere vanishing functions $f_t$; in particular all the contact structures $\ker\alpha_t$ are contactomorphic.

Proof sketch. The argument is the one above, with the deformation replaced by the given family; the Moser equation is solvable at each time because the contact condition is open and the manifold is closed, so the flow exists for all $t \in [0,1]$.

Corollary. A contact structure has no local invariants: in a neighbourhood of every point it is the standard one, and a contact invariant of a manifold is global. This is the contact counterpart of the Darboux theorem for symplectic forms, and it is why the subject is a topology rather than a local geometry.

The Reeb Vector Field

Definition. Let $\alpha$ be a contact form on a manifold of dimension $2n+1$. The Reeb vector field of $\alpha$ is the unique vector field $R_\alpha$ with

$$ \alpha(R_\alpha) = 1, \qquad \iota_{R_\alpha} d\alpha = 0 . $$

Proposition. The Reeb vector field exists and is unique. It is transverse to the contact structure, and it is nowhere vanishing.

Proof. Uniqueness and transversality: if $R$ satisfies the two conditions, then $\alpha(R) = 1$ fixes its component along the contact direction, and $d\alpha(R, v) = 0$ for $v \in \ker\alpha$ fixes its contact-plane component, since $d\alpha$ is nondegenerate on $\ker\alpha$. For existence, pick any vector field $V$ with $\alpha(V) = 1$ and subtract the unique $W \in \ker\alpha$ solving $\iota_W d\alpha = -\iota_V d\alpha$ on $\ker\alpha$; then $R_\alpha = V + W$ satisfies both conditions.

Remark. The Reeb field is a transverse vector field, and together with the contact structure it decomposes the tangent bundle as $TM = \xi\oplus\mathbb{R}R_\alpha$. The contact form is invariant under a contactomorphism only after a conformal factor, so the Reeb field is not determined by the contact structure alone; it changes when the defining form is multiplied by a function. The flow of the Reeb field, the Reeb flow, is a dynamical system, and its periodic orbits, its return map and its asymptotic behaviour are studied in Part III. The contact-geometric input to that study is the field $R_\alpha$ itself; the analysis of its flow is not part of this article.

Example. For the standard form $\alpha_0 = dz + \sum_i x_i\,dy_i$ on $\mathbb{R}^{2n+1}$, the Reeb field is $\partial_z$. For the standard form on the sphere $S^{2n+1}$ it is the Hopf vector field, the generator of the diagonal $U(1)$-action, and its orbits are the Hopf fibres. For the geodesic contact structure on $ST^*Q$ it is the geodesic spray, whose flow is the geodesic flow.

Legendrian Submanifolds

Definition. Let $(M, \xi)$ be a contact manifold of dimension $2n+1$ and let $L \subseteq M$ be a submanifold. Then $L$ is isotropic if $T_xL \subseteq \xi_x$ for every $x \in L$, equivalently if the defining form restricts to zero on $L$, in which case $d\alpha$ restricts to zero on $L$ as well, since $d\alpha|_L = d(\alpha|_L)$; it is Legendrian if it is isotropic and $\dim L = n$.

Proposition. An isotropic submanifold of a contact manifold of dimension $2n+1$ has dimension at most $n$, and a Legendrian submanifold has dimension exactly $n$; the tangent space of a Legendrian at each point is a Lagrangian subspace of the symplectic space $(\xi_x, d\alpha_x)$.

Proof. For $x \in L$ the tangent space $T_xL$ is isotropic for $d\alpha_x$, which is a nondegenerate alternating form on the $2n$-dimensional space $\xi_x$; the dimension bound and the equality $\dim T_xL = n$ for the maximal case are those of the isotropic subspaces of a symplectic vector space, as in Symplectic Forms and Poisson Brackets.

Example. In the standard contact $\mathbb{R}^3$ with $\alpha = dz - y\,dx$, a Legendrian curve is a curve $\gamma(t) = (x(t), y(t), z(t))$ with $\dot z = y\dot x$; the Legendrian condition is a first-order equation on the curve, and it can always be integrated locally. In the standard contact $S^3$ the fibres of the Hopf fibration are the orbits of the Reeb field, hence transverse circles, while the standard Legendrians are the curves tangent to the contact planes; the classification of both families is part of Symplectic and Contact Topology.

Example. In the jet space $J^1(N)$ with $\alpha = dz - \sum_i p_i\,dq^i$, the $1$-jet of a function and the graph of a closed $1$-form are Legendrian, as noted above. Conversely, a Legendrian submanifold of $J^1(N)$ that is transverse to the fibres of $J^1(N)\to N\times\mathbb{R}$ is locally the $1$-jet graph of a function. This is the contact analogue of the Lagrangian neighbourhood theorem, and it is the reason the jet space is the local model for arbitrary Legendrian submanifolds: a neighbourhood of a Legendrian in a contact manifold is contactomorphic to a neighbourhood of the zero section in a jet space.

Symplectisation

The passage from contact geometry to symplectic geometry is the symplectisation.

Definition. Let $(M, \alpha)$ be a cooriented contact manifold of dimension $2n+1$. The symplectisation is the manifold $M\times\mathbb{R}$ with coordinate $t$ in the second factor and the $2$-form

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

Proposition. The form $\omega = d(e^t\alpha)$ is symplectic on $M\times\mathbb{R}$.

Proof. It is closed because it is exact. For nondegeneracy, compute the top power:

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

where all terms containing a repeated factor among $dt$, $\alpha$ and $d\alpha$ vanish except the displayed one: the only surviving term is the one with one factor $dt\wedge\alpha$ and $n$ factors of $d\alpha$, whose coefficient is $\binom{n+1}{1} = n+1$, and $\alpha\wedge(d\alpha)^{\wedge n}\neq 0$ by the contact condition. Hence the top power is nowhere vanishing and $\omega$ is nondegenerate.

Remark. The symplectisation is the precise sense in which a contact manifold is the boundary at infinity of a symplectic manifold: the level sets $M\times\{t\}$ are contact-type hypersurfaces, the restriction of $\omega$ to each is $e^t d\alpha$, and the contact form is recovered as the contraction of $\omega$ with a suitable transverse field. Conversely, a hypersurface in a symplectic manifold along which the symplectic form has this shape is said to be of contact type, and the contact geometry is the study of the boundary. Legendrian submanifolds of $M$ lift to Lagrangian submanifolds of the symplectisation by $L \mapsto L\times\mathbb{R}$, which is the geometric content of the correspondence: $\omega$ restricts to zero on $L\times\mathbb{R}$ because it restricts to zero on $L$ and $dt$ pairs with $\alpha$ only through the transverse direction.

Example. The symplectisation of the standard contact sphere $S^{2n+1}$ is $\mathbb{C}^{n+1}\setminus\{0\}$ with the standard symplectic form, and the contact form is the restriction of the Liouville form to the spheres of each radius. The unit sphere is the boundary of the unit ball, which is a Stein doma, and the contact structure it inherits is the standard one. This is the model of a symplectic filling, and the existence and classification of fillings is a central problem.

Contact Structures from Complex Geometry

Definition. Let $X$ be a complex manifold of complex dimension $n+1$, let $\rho : X \to \mathbb{R}$ be a smooth function and let $\Sigma = \rho^{-1}(0)$ be a regular level set. The hypersurface $\Sigma$ is strictly pseudoconvex if the Levi form of $\rho$ is positive definite on the complex tangent space $T\Sigma\cap i\,T\Sigma$ at each point.

Proposition. The complex tangent space of a hypersurface in a complex manifold is a hyperplane field of real codimension one, and a strictly pseudoconvex hypersurface carries a natural contact form: the restriction to $\Sigma$ of $\alpha = -d^c\rho$, where $d^c$ is the twisted differential. The resulting contact structure has $T\Sigma\cap i\,T\Sigma$ as its contact planes.

Proof sketch. The complex tangent bundle $T\Sigma\cap i\,T\Sigma$ has real dimension $2n$ in $\Sigma$ of dimension $2n+1$, so it is a hyperplane field. The Levi form is, up to a factor, the restriction of $d\alpha$ to the complex tangent space, and strict pseudoconvexity is the statement that this restriction is definite, hence nondegenerate; the contact condition follows.

Example. The unit sphere $S^{2n+1}$ in $\mathbb{C}^{n+1}$ is the regular level set of $\rho(z) = |z|^2 - 1$, and it is strictly pseudoconvex; the induced contact structure is the standard one of the examples above. The general construction produces a contact structure on the boundary of every strictly pseudoconvex doma, and in dimension three the boundaries of the domains in $\mathbb{C}^2$ are the standard source of contact $3$-manifolds. The complex and Kähler structures used here are those andbeing; a contact metric structure compatible with the contact form in this way is a Sasakian structure, and the Sasakian manifolds form the odd-dimensional companion family to the Kähler manifolds.

Summary

A contact manifold is a manifold of odd dimension $2n+1$ carrying a hyperplane field $\xi = \ker\alpha$ defined by a $1$-form $\alpha$ with $\alpha\wedge(d\alpha)^{\wedge n}\neq 0$. The field is maximally non-integrable: the integral manifolds of $\xi$ have dimension at most $n$, and the maximal ones are the Legendrian submanifolds. The contact structure is the field, not the form; a defining form is unique up to a nowhere vanishing factor $f$, under which $\alpha\wedge(d\alpha)^{\wedge n}$ scales by $f^{\,n+1}$, and a cooriented contact manifold is oriented by that volume form.

The standard examples are $\mathbb{R}^{2n+1}$ with $\alpha_0 = dz + \sum_i x_i\,dy_i$, the sphere $S^{2n+1}$ with the restriction of the radial form, the spherical cotangent bundle $ST^*Q$ with the geodesic contact structure, and the jet space $J^1(N)$ with $\alpha = dz - \sum_i p_i\,dq^i$, in which the $1$-jets of functions are Legendrian. The Darboux theorem puts every contact form locally into the standard shape, and Gray's theorem shows that a deformation of contact forms on a closed manifold is realised by an isotopy; so a contact structure has no local invariants. Every contact form has a unique Reeb vector field $R_\alpha$ with $\alpha(R_\alpha) = 1$ and $\iota_{R_\alpha}d\alpha = 0$, transverse to the contact planes.

Legendrian submanifolds are the isotropic submanifolds of dimension $n$, and a neighbourhood of a Legendrian is modelled on a neighbourhood of the zero section of a jet space. The symplectisation $M\times\mathbb{R}$ with $\omega = d(e^t\alpha)$ is a symplectic manifold whose level sets are contact-type hypersurfaces, and Legendrians lift to Lagrangians; conversely, a hypersurface of contact type in a symplectic manifold inherits a contact structure. Finally, a strictly pseudoconvex hypersurface in a complex manifold is contact, the unit sphere in $\mathbb{C}^{n+1}$ being the model, and the compatible metric structures are the Sasakian ones.

Summary of Notation

Symbol	Meaning
$M$, $\dim M = 2n+1$	Odd-dimensional contact manifold; $n$ the contact rank
$\alpha$	Contact form, $\alpha\wedge(d\alpha)^{\wedge n}\neq 0$
$\xi = \ker\alpha$	Contact structure, the maximally non-integrable hyperplane field
$\alpha_0 = dz + \sum_i x_i\,dy_i$	Standard contact form on $\mathbb{R}^{2n+1}$
$d\alpha\vert_\xi$	Nondegenerate alternating form on each contact plane
$f\alpha$, $f^{\,n+1}$	Rescaling of a contact form and its effect on $\alpha\wedge(d\alpha)^{\wedge n}$
Contactomorphism $F$	$dF(\xi) = \eta$; $F^*\alpha_N = f\alpha_M$ in the cooriented case
$R_\alpha$	Reeb vector field, $\alpha(R_\alpha)=1$, $\iota_{R_\alpha}d\alpha=0$
$TM = \xi\oplus\mathbb{R}R_\alpha$	Splitting by the Reeb field
$J^1(N) = T^*N\times\mathbb{R}$	Jet space, $\alpha = dz - \sum_i p_i\,dq^i$; $j^1f$ Legendrian
Isotropic, Legendrian	$TL\subseteq\xi$; isotropic of dimension $n$
$M\times\mathbb{R}$, $\omega = d(e^t\alpha)$	Symplectisation; contact manifold as a contact-type boundary
$\Sigma = \rho^{-1}(0)$ strictly pseudoconvex	Hypersurface in a complex manifold; carries a contact form $-d^c\rho\vert_\Sigma$
Sasakian structure	Contact structure compatible with a metric, odd-dimensional analogue of Kähler