Remark 3.5 . [05DQ]

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Remark 3.5.

The coordinates $x_{1},\cdots,x_{n}$ on $\bar{S}$ in the proof of this proposition induce parallel 1-forms $dx_{1},\cdots,dx_{n}$ on $(\tilde{S},h)$ £¬ which are pointwise linear independent, i.e. $dx_{1},\cdots,dx_{n}$ is a global parallel frame field. Under the coordinates $y_{1},\cdots,y_{n}$ on $\bar{S}^{\perp}$ , we have these formulas

\pi^{*}g_{0}=\sum(dx_{j}^{2}+dy_{j}^{2}),\ \ \ \pi^{*}\omega_{0}=\sum dx_{j}\wedge dy_{j},\ \ \ e^{\sqrt{-1}\theta_{0}}\pi^{*}\Omega_{0}=\bigwedge_{j=1}^{n}(dx_{j}+\sqrt{-1}dy_{j}).