## definition 0.0.1

Let \(M\) be a closed \(2n+1\)-real dimensional manifold. A*contact form*on \(M\) is one form \(\alpha\in \Omega^1(M)\) so that \[\alpha \wedge (d\alpha)^n \] is a volume form on \(M\). We call the pair \((M, \alpha)\) a contact manifold.

## example 0.0.2

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.## definition 0.0.4

Let \((M, \alpha)\) be a contact manifold. The Reeb vector field is the unique vector field \(R_\alpha\) characterized by \begin{align*} R_\alpha\in \ker(d\alpha) && \alpha(R_\alpha)=1.\end{align*}## example 0.0.5

We return to example 0.0.2 of hypersurfaces in \(\RR^{2n}\) given by \(M=p^{-1}(N)\). Consider the hypersurface \(N\) defined by the equation \(p_1+\cdots p_n=1\). Then \(M=S^{2n-1}\subset \RR^n\). We give \(\CC^n\) the polar coordinates \((r_i, \theta_i)\). Let \(f=\sum_{i=1}^n |z_i|^2.\) The tangent space to \(M\) is the orthogonal complement to \(\grad(f).\) Consider the vector field \[V:=\sum_{i=1}^n x_i \partial_{y_i}- y_i \partial_{x_i}.\] First, observe that \[V\cdot \grad(f)=\sum_{i=1}^n( x_i y_i - y_i x_i )= 0 \] so \(V\) restricts to a vector field on \(M\). Let \(v=\sum_{i}a_i \partial_{x_i}+b_i\partial_{y_i}\) be any vector in \(TM\). Then the pairing \begin{align*} \omega(v, V)=&\sum_{i=1}^n a_ix_i+b_iy_i\\ =&v\cdot \grad(f)=0 \end{align*} From this, we conclude that \(V\in \ker(d\alpha)\). Finally, we have that \(\alpha=\iota_Z\omega\), so \[\omega(Z, V)=\sum_{i=1}^n (x^2+y^2)=1\] From which we conclude that \(V\) is the Reeb vector field for \((M, \alpha)\).## example 0.0.6

We return to example 0.0.5 of the Reeb vector field on \((S^{n-1},\alpha)\), where \(S^{n-1}\) is considered as a hypersurface of \(\CC^n\). Recall we have a map \(p:S^{2n-1}\to N\), where \(N\subset \RR_{\geq 0}^n\) is the simplex defined by \(\sum_{i=1}^n p_i^2=1\). The fibers of \(p\) are \(n\)-dimensional tori in \(S^{2n-1}\) which are parallel to the Reeb vector field \(V_\alpha\). In fact, Reeb vector field \(V_\alpha\) acts on the fibers \(p^{-1}(p_1, \ldots, p_n)\) by translation in the \((\sqrt p_1, \ldots, \sqrt p_n)\) direction. We therefore identify two types of fibers of \(p\):- If \((\sqrt p_1, \ldots, \sqrt p_n)\) has integral slope (that is, there exists a scalar \(r\) so that \(r\cdot(\sqrt p_1, \ldots, \sqrt p_n) \in \ZZ^n\)) then every point on the fiber belongs to a closed orbit.
- Otherwise, no point on the fiber belongs to a closed orbit.

## proposition 0.0.7

Let \(M\) be a contact manifold. Let \[\Gamma:=\{\gamma: \RR/\ell\RR\to M \st\gamma'=R\}\] denote the set of Reeb orbits. The image of \(\ell(\Gamma)\subset \RR_{\geq 0}\) is countable and closed.## definition 0.0.8

Let \((M, \alpha)\) be a contact manifold. The symplectization of \((M, \alpha)\) is the symplectic manifold \[(\RR\times M, d(\exp(r)\cdot \alpha)).\]## definition 0.0.9

Let \(M\) be a contact manifold, and let \(\RR\times M\) be its symplectization. A*linear Hamiltonian*of slope \(m\) is the Hamiltonian \(H^m(r, x)= m\cdot \exp(r)\).