contact metric structure on squashed spheres

Question

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.

Answer 1

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 $X, Y$.

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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 $i$.