Let $M$ be a smooth manifold and $\alpha$ a one form on $M$ Then $\alpha$ is a contact form on $M$ if

1. for each point $m \in M$ $\alpha_{m} \neq 0$ and
2. the restriction $d\alpha_{m}|_{\ker{\alpha_{m}}}$ of the differential of $\alpha$ is nondegenerate.

Condition 1 ensures that $\xi=\ker{\alpha}$ is a subbundle of the vector bundle $TM$ Condition 2 equivalently says $d\alpha$ is a symplectic structure on the vector bundle $\xi \to M$ A contact structure $\xi$ on a manifold $M$ is a subbundle of $TM$ so that for each $m \in M$ there is a contact form $\alpha$ defined on some neighborhood of $m$ so that $\xi = \ker{\alpha}$ A co-oriented contact structure is a subbundle of $TM$ of the form $\xi=\ker{\alpha}$ for some globally defined contact form $\alpha$

A (co-oriented) contact manifold is a pair $(M,\xi)$ where $M$ is a manifold and $\xi$ is a (co-oriented) contact structure. Note, symplectic linear algebra implies that $\dim{M}$ is odd. If $\dim{M} = 2n+1$ for some positive integer $n$ then a one form $\alpha$ is a contact form if and only if $\alpha \wedge (d\alpha)^{n}$ is everywhere nonzero.

Suppose now that $(M_1,\xi_1=\ker{\alpha_1})$ and $(M_2,\xi_2=\ker{\alpha_2})$ are co-oriented contact manifolds. A diffeomorphism $\phi:M_1 \to M_2$ is called a contactomorphism if the pullback along $\phi$ of $\alpha_2$ differs from $\alpha_1$ by some positive smooth function $f:M_1 \to \mathbb{R}$ that is, $\phi^{*}\alpha_{2} = f\alpha_{1}$

Examples:

1. $\mathbb{R}^3$ is a contact manifold with the contact structure induced by the one form $\alpha = dz + xdy$
2. Denote by $\mathbb{T}^{2}$ the two-torus $\mathbb{T}^{2} = S^1 \times S^1$ Then, $\mathbb{R} \times \mathbb{T}^2$ (with coordinates $t,\theta_1,\theta_2$ is a contact manifold with the contact structure induced by $\alpha = \cos{t\theta_1}+\sin{t\theta_2}$