A key set of examples of contact manifolds come as hypersurfaces of symplectic manifolds. Let \((X, \omega)\) be a symplectic manifold. Suppose that there is an expanding vector field \(Z\) on \(X\), that is, a vector field so that
\[\mathcal L_Z \omega = \omega.\]
The symplectic manifold \(X\) is exact, with primitive given by \(\lambda=\iota_Z \omega\).
Let \(i:M\into X\) be a hypersurface which is transverse to \(Z\). Then the restriction \(\alpha:=\lambda|_M\) is an example of a contact form. We see that the form
\begin{align*}
\alpha \wedge d\alpha^{n-1} =& i^* (\iota_Z \omega \wedge \omega^{n-1})\\
\end{align*}
is nonvanishing, as \(\omega^n\) is a volume form and \(Z\) is transverse to \(M\).
The simplest example to consider come from \(\CC^n= \RR^{2n}\), where the radial vector field \(Z=\frac{1}{2}\sum_i \left(x_i \partial_{x_i} + y_i\partial_{y_i}\right)\) provides an example of an expanding vector field. The associated primitive for the symplectic form is
\[\iota_Z \omega =\sum_{i=1}^n x_i dy_i-y_idx_i \]
This radial vector field plays especially nicely with respect to the moment map,
\begin{align*}
p: \RR^{2n}\to& (\RR_{\geq 0})^n\\
(x_i, y_i)\mapsto&\frac{1}{2} (x_i^2+y_i^2)
\end{align*}
We give the base of the moment polytope \((\RR_{\geq 0})^n\) coordinates \((p_1, \ldots, p_n\)).
At every point \((x_i, y_i)\in \RR^{2n}\), we can project the Liouville vector field to
\[p_*Z_{(x_i, y_i)}= \sum_{i=1}^n (x_i^2+y_i^2) \partial_{p_i} =\frac{1}{2}\sum_{i=1}^n p_i \partial_{p_i}.\]
In particular, if we have a hypersurface \(N\subset (\RR_{\geq 0})^n\) which is transverse to the radial vector field \(\sum_{i=1}^n p_i \partial_{p_i}\) and whose preimage \(M:=p^{-1}(N)\subset \RR^{2n}\) is a smooth hypersurface, then \(M\) is a contact manifold.
figure 0.0.2:The Liouville structure on \(\CC^2\) as viewed from the moment map
