# 数学代写|拓扑学代写Topology代考|Contact submanifolds

#### Doug I. Jones

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## 数学代写|拓扑学代写Topology代考|Contact submanifolds

Let $\left(M^{\prime}, \xi^{\prime}=\operatorname{ker} \alpha^{\prime}\right) \subset(M, \xi=\operatorname{ker} \alpha)$ be a contact submanifold, that is, $\left.T M^{\prime} \cap \xi\right|{M^{\prime}}=\xi^{\prime}$. As before we write $\left.\left(\xi^{\prime}\right)^{\perp} \subset \xi\right|{M^{\prime}}$ for the symplectically orthogonal complement of $\xi^{\prime}$ in $\left.\xi\right|{M^{\prime}}$. Since $M^{\prime}$ is a contact submanifold (so $\xi^{\prime}$ is a symplectic sub-bundle of $\left.\left(\left.\xi\right|{M^{\prime}}, d \alpha\right)\right)$, we have
$$T M^{\prime} \oplus\left(\xi^{\prime}\right)^{\perp}=\left.T M\right|_{M^{\prime}},$$
i.e. we can identify $\left(\xi^{\prime}\right)^{\perp}$ with the normal bundle $N M^{\prime}$. Moreover, $d \alpha$ induces a conformal symplectic structure on $\left(\xi^{\prime}\right)^{\perp}$, see Lemma 1.3.4.
Definition 2.5.14 The bundle
$$\operatorname{CSN}_M\left(M^{\prime}\right):=\left(\xi^{\prime}\right)^{\perp}$$
with the conformal symplectic structure induced by $d \alpha$ is called the conformal symplectic normal bundle of $M^{\prime}$ in $M$.

Theorem 2.5.15 Let $\left(M_i, \xi_i\right), i=0,1$, be contact manifolds with compact contact submanifolds $\left(M_i^{\prime}, \xi_i^{\prime}\right)$. Suppose there is an isomorphism of conformal symplectic normal bundles $\Phi: \operatorname{CSN}{M_0}\left(M_0^{\prime}\right) \rightarrow \operatorname{CSN}{M_1}\left(M_1^{\prime}\right)$ that covers a contactomorphism $\phi:\left(M_0^{\prime}, \xi_0^{\prime}\right) \rightarrow\left(M_1^{\prime}, \xi_1^{\prime}\right)$. Then $\phi$ extends to a contactomorphism $\psi$ of suitable neighbourhoods $\mathcal{N}\left(M_i^{\prime}\right)$ of $M_i^{\prime}$ such that $\left.T \psi\right|_{\operatorname{CSN}\left(M_0, M_0^{\prime}\right)}$ and $\Phi$ are bundle homotopic (as conformal symplectic bundle isomorphisms).

## 数学代写|拓扑学代写Topology代考|Hypersurfaces

Let $S$ be an oriented hypersurface in a contact manifold $(M, \xi=\operatorname{ker} \alpha)$ of dimension $2 n+1$. In a neighbourhood of $S$ in $M$, which we can identify with $S \times \mathbb{R}$ (and $S$ with $S \times{0}$ ), the contact form $\alpha$ can be written as
$$\alpha=\beta_r+u_r d r$$
where $\beta_r, r \in \mathbb{R}$, is a smooth family of 1 -forms on $S$ and $u_r: S \rightarrow \mathbb{R}$ a smooth family of functions. The contact condition $\alpha \wedge(d \alpha)^n \neq 0$ then becomes, with the derivative with respect to $r$ denoted by a dot,
\begin{aligned} 0 & \neq \alpha \wedge(d \alpha)^n \ & =\left(\beta_r+u_r d r\right) \wedge\left(d \beta_r-\dot{\beta}r \wedge d r+d u_r \wedge d r\right)^n \ & =\left(-n \beta_r \wedge \dot{\beta}_r+n \beta_r \wedge d u_r+u_r d \beta_r\right) \wedge\left(d \beta_r\right)^{n-1} \wedge d r . \end{aligned} The intersection $T S \cap\left(\left.\xi\right|_S\right)$ determines a distribution (of non-constant rank) of subspaces of $T S$. If $\alpha$ is written as above, this distribution is given by the kernel of $\beta_0$, and hence, at a given $p \in S$, defines either the full tangent space $T_p S$ (if $\beta{0, p}=0$ ) or a 1-codimensional subspace both of $T_p S$ and $\xi_p$ (if $\beta_{0, p} \neq 0$ ). In the former case, the symplectically orthogonal complement $\left(T_p S \cap \xi_p\right)^{\perp}$ (with respect to the conformal symplectic structure $d \alpha$ on $\xi_p$ ) is ${\mathbf{0}}$; in the latter case, $\left(T_p S \cap \xi_p\right)^{\perp}$ is a 1 -dimensional subspace of $\xi_p$ contained in $T_p S \cap \xi_p$.

From that it is intuitively clear what one should mean by a ‘singular 1-dimensional foliation’, and we make the following somewhat provisional definition, where the hypersurface $S$ need not be orientable.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Contact submanifolds

$$T M^{\prime} \oplus\left(\xi^{\prime}\right)^{\perp}=\left.T M\right|_{M^{\prime}},$$

2.5.14 bundle
$$\operatorname{CSN}_M\left(M^{\prime}\right):=\left(\xi^{\prime}\right)^{\perp}$$

## 数学代写|拓扑学代写Topology代考|Hypersurfaces

$$\alpha=\beta_r+u_r d r$$

\begin{aligned} 0 & \neq \alpha \wedge(d \alpha)^n \ & =\left(\beta_r+u_r d r\right) \wedge\left(d \beta_r-\dot{\beta}r \wedge d r+d u_r \wedge d r\right)^n \ & =\left(-n \beta_r \wedge \dot{\beta}r+n \beta_r \wedge d u_r+u_r d \beta_r\right) \wedge\left(d \beta_r\right)^{n-1} \wedge d r . \end{aligned}交集$T S \cap\left(\left.\xi\right|_S\right)$决定了$T S$的子空间的分布(秩非恒定的)。如果$\alpha$如上所述，则该分布由$\beta_0$的内核给出，因此，在给定的$p \in S$上，定义了完整的切空间$T_p S$(如果$\beta{0, p}=0$)或$T_p S$和$\xi_p$的1余维子空间(如果$\beta{0, p} \neq 0$)。在前一种情况下，辛正交补$\left(T_p S \cap \xi_p\right)^{\perp}$(关于$\xi_p$上的共形辛结构$d \alpha$)为${\mathbf{0}}$;在后一种情况下，$\left(T_p S \cap \xi_p\right)^{\perp}$是包含在$T_p S \cap \xi_p$中的$\xi_p$的一维子空间。

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