My goal to write down an explicit (and simplest) contact metric structure on squashed $S\_\omega^{2n + 1}$ defined as
\begin{equation}
 S\_\omega ^{2n + 1} = \left\{ {\left( {{z\_i}} \right) \in \mathbb{C}^{n + 1}\;\;|\sum\limits\_{i = 1}^{n + 1} {\omega \_i^2{{\left| {{z\_i}} \right|}^2} = 1} } \right\}
\end{equation}


For the contact 1-form I would use the natural one restricted on $S^{2n+1}\_\omega$
\begin{equation}
 \kappa \equiv \frac{i}{2}\sum\limits\_{i = 1} {\left( {{z\_i}d{{\bar z}\_i} - {{\bar z}\_i}d{z\_i}} \right)} 
\end{equation}


Take $n = 2$ as example, I use parametrization
\begin{equation}
 \left\{ \begin{gathered}
 {z\_1} = \omega \_1^{ - 1}\sin {\rho \_1}\cos {\rho \_2}{e^{i{\varphi \_1}}} \\
 {z\_2} = \omega \_2^{ - 1}\sin {\rho \_1}\sin {\rho \_2}{e^{i{\varphi \_2}}} \\
 {z\_3} = \omega \_3^{ - 1}\cos {\rho \_1}{e^{i{\varphi \_3}}} \\ 
\end{gathered} \right.
\end{equation}


and if I am calculating correctly, the Reeb vector field will be
\begin{equation}
 R = \sum\limits\_i {\omega \_i^2{\partial \_{{\varphi \_i}}}} 
\end{equation}
which generates rotation $\left( {{z\_1},{z\_2},{z\_2}} \right) \to \left( {{e^{i\omega \_1^2\theta }}{z\_1},{e^{i\omega \_2^2\theta }}{z\_2},{e^{i\omega \_3^2\theta }}{z\_2}} \right)$ (Since that $\kappa \sim zd\bar z$ has $\omega\_i^{-2}$ coefficients, so $R$ will need $\omega\_i^{2}$ to kill these ugly coefficients and leave some $\sin$ and $\cos$ so that $\iota\_R \kappa$ to add up to 1).



> 
> But I am stuck at finding the compatible metric $g$, and $\Phi$ that $\Phi^2 = -1 + \kappa \otimes R$.
> 
> 
> 


I naively guessed the metric induced from $\sum\limits\_i {d{z\_i}d{{\bar z}\_i}} $, but it does not satisfy $g\_{mn}R^m = \kappa\_n$, since the power of $\omega\_i$ does not match: $g\_{mn} \sim \omega^{-2}\_i$, which cannot do the job. I tried also $\sum\limits\_k {\omega \_i^{ - 2}d{z\_i}d{{\bar z}\_i}} $, but then ${\Phi ^m}\_n \equiv {g^{mk}}{\left( {d\kappa } \right)\_{kn}}$ does not seem to give $\Phi^2 \sim -1 + R\kappa$.



> 
> Any comments on how to find the $g$ and $\Phi$? Reference will be great.
> 
> 
> 



contact metric structure on squashed spheres
After skimming through Section 2 and especially Subsection 2.3 of the paper by C. Boyer: <http://arxiv.org/abs/1003.1903> it seems to me that you have to choose a compatible almost complex structure $J$ on your contact manifold $(S^{2n+1}\_{\omega}, \mathcal{D}=\text{ker}\,\kappa)$ and set $$\Phi(X) = JX \text{ for } X \in \mathcal{D} \quad\text{ and }\quad \Phi(R) = 0,$$


i.e. on the contact distribution $\mathcal{D}$ the map $\Phi$ agrees with $J$ while in the Reeb direction $\Phi$ acts as the zero operator.


A compatible Riemannian metric is then defined by $$ g(X, Y) = d\kappa(\Phi(X), Y) + \kappa(X)\cdot \kappa(Y)$$ for any tangent vectors $X, Y$.




---


I think in your case of the squashed spheres you can take $J$ to be the almost complex structure induced from the standard complex structure on $\mathbb{C}^{n+1}$ --- so $J$ is just multiplication by $i$.


