= Contact topology
{title2=$\xi=\ker\alpha$}
<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>.<Lectures on Symplectic Geometry by Ana Cannas da Silva>[sections 10.1–10.3]{ref}
In $\mathbb R^3$, take $\alpha=dz-y\,dx$. Then $d\alpha=dx\wedge dy$ and $\alpha\wedge d\alpha=dz\wedge dx\wedge dy\ne0$. 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>.
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