Contact Manifold

Contact Element

A contact element on a manifold M is a point $p\in M$, called the contact point, together with a tangent hyperplane at $p$, $H_p\subset T_{p}M$, that is, a codimension-1 subspace of $T_{p}M$.

A hyperplane $H_p\subset T_{p}M$ determines a covector ${\alpha }_p\in T_p^{\ast }M\setminus \{0\}$, up to multiplication by a nonzero scalar:

$$(p,H_p)\text{ is a contact element}⟷H_p=\ker {\alpha }_p\text{ with }{\alpha }_p:T_{p}M⟶\mathbb{R}\text{ linear },\ne 0$$ $$\ker {\alpha }_p=\ker {\alpha }_p^{\prime }⟷{\alpha }_p=\lambda {\alpha }_p^{\prime }\text{ for some }\lambda \in \mathbb{R}\setminus \{0\}.$$

Contact Structure

Suppose that $H$ is a smooth field of contact elements (i.e., of tangent hyperplanes) on $M$:

H : p \longmapsto H_p \subset T_p M ;.

Locally, $H = \ker \alpha$ for some 1-form $\alpha$, called a locally defining 1-form for $H$. ($\alpha$ is not unique: $\ker \alpha = \ker(f \alpha)$, for any nowhere vanishing $f : M \to \mathbb{R}$.) A contact structure on $M$ is a smooth field of tangent hyperplanes $H \subset TM$, such that, for any locally defining 1-form $\alpha$, we have $d\alpha|_H$ non-degenerate (i.e., symplectic). The pair $(M,H)$ is then called a contact manifold and $\alpha$ is called a local contact form.

Remark

At each $p\in M$, $T_{p}M=\underset{H_p}{\underbrace{\ker {\alpha }_p}}\oplus \underset{\text{1-dimensional}}{\underbrace{\ker d{\alpha }_p}}.$ Let $(M,H)$ be a contact manifold. A global contact form exists if and only if the quotient line bundle $TM/H$ is orientable. Since $H$ is also orientable, this implies that $M$ is orientable.

Proposition

Let $H$ be a field of tangent hyperplanes on $M$. Then $H\text{ is a contact structure}⟺\alpha \wedge (d\alpha {)}^n\ne 0\text{ for every locally defining 1-form }\alpha .$

Theorem

Let $(M,H)$ be a contact manifold and $p\in M$. Then there exists a coordinate system $(\mathcal{U},x_1,y_1,\dots ,x_n,y_n,z)$ centered at $p$ such that on $\mathcal{U}$

\alpha = \sum x_i dy_i + dz \text{ is a local contact form for } H ;.

>[!theorem] Gray Theorem > >Let $M$ be a compact manifold. Suppose that $\alpha_t, t \in [0, 1]$, is a smooth family of (global) contact forms on $M$. Let $H_t = \ker \alpha_t$. Then there exists an isotopy $\rho : M \times \mathbb{R} \longrightarrow M$ such that $H_t = \rho_{t*}H_0$, for all $0 \le t \le 1$.