\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
We return to (contact hypersurfaces) 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 \((M, \alpha)\).