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Contact topology studies global properties of contact structures on odd-dimensional smooth manifolds. Locally such a structure is a hyperplane distribution $\xi =\ker \alpha$ with $\alpha \wedge (d\alpha {)}^n$ nowhere zero in dimension $2n+1$. Multiplying $\alpha$ by a nowhere-zero function leaves the hyperplane distribution unchanged: the structure is the field of hyperplanes, not a preferred one-form.^[ref]

In ${\mathbb{R}}^3$, take $\alpha =dz-ydx$. Then $d\alpha =dx\wedge dy$ and $\alpha \wedge d\alpha =dz\wedge dx\wedge dy\ne 0$. The spanning vector fields ${\partial }_x+y{\partial }_z$ and ${\partial }_y$ have Lie bracket $-{\partial }_z$, outside the distribution. Thus the planes cannot be tangent to a foliation by surfaces. The local normal form is universal, while global topology distinguishes contact structures and constrains their relation to symplectic geometry.

The differential-forms version of the Frobenius theorem says that a constant-rank distribution $D={\bigcap }_i\ker {\alpha }^i$ is integrable exactly whenfor suitable 1-forms ${\beta }^i{}_j$. For a plane distribution $D=\ker \alpha$ on ${\mathbb{R}}^3$, this reduces to the integrability criterion for a plane distribution $\alpha \wedge d\alpha =0$.

For example, $\ker dz$ is integrable: its integral surfaces are the horizontal planes $z=\text{constant}$. In contrast, for $\alpha =dz-xdy$,Thus $\ker (dz-xdy)$ is not integrable; it is the standard contact structure on ${\mathbb{R}}^3$.

A Stein surface is a Stein manifold of complex dimension two, hence real dimension four. Its global holomorphic functions separate points and provide local coordinates, and the holomorphic hull of every compact subset is compact. The word “surface” here refers to complex dimension, rather than to a real two-dimensional surface.

The affine complex plane ${\mathbb{C}}^2$ is a Stein surface, as is the smooth closed hypersurface $xy-z^2=1$ in ${\mathbb{C}}^3$: its gradient $(y,x,-2z)$ never vanishes on the hypersurface. A compact connected positive-dimensional complex manifold cannot be Stein, because its holomorphic functions are constant. In real dimension four, the topology of Stein surfaces has especially rich links with handle decompositions and contact structures on three-dimensional boundaries.^[ref]