\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
We return to (contact hypersurfaces) of 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.
The Reeb orbits of \((S^{n-1}, \alpha)\) are in bijection with \(\bigoplus_{\vec v\in \NN^n\setminus \{0\}} T^n/\vec v\).