Contact structure

A contact structure on a three-manifold is a plane distribution locally of the form $\ker \alpha$ with $\alpha \wedge d\alpha$ nowhere zero. It is maximally nonintegrable; the standard example on ${\mathbb{R}}^3$ is $\ker (dz-xdy)$.

A smooth interval family of contact forms on a compact manifold without boundary is carried back to its initial contact distribution by a smooth isotopy. The pullback of each form is a positive smooth multiple of the initial form. Solve for the horizontal generator of contact stability and integrate the resulting scalar equation for the contact conformal factor along an isotopy.

If a time-dependent vector field satisfies ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, its flow maps satisfy ${\rho }_t^{\ast }{\alpha }_t=u_t{\alpha }_0$, where $u_t(x)=\exp ({\int }_0^t{h}_s({\rho }_s(x))ds)$. Thus the conformal factor is smooth and positive even when $h_t$ changes sign.

For a family of contact forms, require $Y_t\in \ker {\alpha }_t$ and solve $({\iota }_{Y_t}d{\alpha }_t){|}_{\ker {\alpha }_t}=-{\dot{\alpha }}_t{|}_{\ker {\alpha }_t}$. The contact distribution symplectic form gives a unique smooth solution. Then ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, with $h_t={\dot{\alpha }}_t(R_t)$ and $R_t$ the Reeb vector field.

A contact form $\alpha$ defines the hyperplane distribution $\xi =\ker \alpha$. Multiplying $\alpha$ by a nowhere-zero function leaves this distribution unchanged. The contact distribution symplectic form distinguishes it from an integrable hyperplane distribution.

For a contact form on a $(2r+1)$-manifold, $d\alpha {|}_{\ker \alpha }$ is nondegenerate: the contact condition says its top exterior power is nowhere zero on the hyperplanes. This fiberwise symplectic form identifies contact-plane vectors with covectors on that plane and determines the horizontal generator of contact stability.

A contact form on a $(2n+1)$-manifold is a one-form satisfying $\alpha \wedge (d\alpha {)}^n\ne 0$ everywhere.

The Reeb vector field of a contact form is uniquely characterized by

For ${\alpha }_s=\frac{1}{2}(r_1^{2}d{\theta }_1+s{r}_2^{2}d{\theta }_2)$ on $S^3$, the Reeb field is $2{\partial }_{{\theta }_1}+2s^{-1}{\partial }_{{\theta }_2}$. If $s$ is irrational, its only closed orbits are the two coordinate circles.