Integrability of an "almost" CR-structure defined thanks to an almost contact metric structure.

Question

Let $(M,\eta,\xi,\varphi)$ be an almost contact metric manifold, that is:

• $M$ is a smooth manifold
• $\eta$ is a contact 1-form on $M$, i.e $\eta$ is a 1-form and $\mathrm{d}\eta|_{\ker \eta}$ is non-degenerate
• $\xi$ is the Reeb vector field of $\eta$, that is $\xi \in \ker \mathrm{d}\eta$ and $\eta(\xi) \equiv 1$
• $\varphi$ is a section of $End(TM)$ with $\varphi^2 = -\mathrm{id} + \eta\otimes\xi$ and $\varphi (\xi) = 0$

One can think of this manifold as an "almost" (I don't know if this is the right terminology) CR manifold: let $H = \ker \eta$ be the contact distribution and $J = \varphi|_{H}$, which is a section of $End(H)$. Then $J$ is a complex structure on the contact distribution $H$. For $(M,H,J)$ to be a CR manifold, $J$ needs to be integrable on $H$, that is, in the complexified bundle $(H^{\mathbb{C}},J)$ with the usual holomorphic/anti-holomorphic decomposition $H^{\mathbb{C}} = H^{1,0} \oplus H^{0,1}$, one needs to have $[H^{1,0},H^{1,0}] \subset H^{1,0}$.

I remember having read a paper where it was stated that such an "almost" CR structure is integrable and leads to a CR manifold if and only if the Nijenhuis tensor of $\varphi$, defined by $$ N(X,Y) = -[\varphi X,\varphi Y]+ \varphi[\varphi X,Y] + \varphi[X,\varphi Y] - \varphi^2[X,Y] $$ satisfies a certain condition. I think this condition was something like $N$ taking values proportional to $\xi$.

The problem is I cannot find this paper anymore, and some long research on the Internet and on arXiv had been unproductive. Moreover, I remember that, in this paper, this was only stated and wasn't proved.

My question is the following: does anybody have a reference about this fact?

Edit

Here are some more lines to be more precise about my question. Regarding only $J$, it is clear that $(M,H,J)$ is integrable if and only if the Nijenhuis tensor of $J$ indentically vanishes. My question is more about how to translate this condition to a condition on the full tensor $\varphi$ and not just its restriction?

A - possibly abusive - parallel can be done with almost Kaehler geometry: given an almost Kaehler manifold $(M,g,J)$, there are three equivalent conditions that assure the integrability of the structure, that is $(M,g,J)$ is a Kaehler manifold. It may happen that some condition is more relevant in some context than the others: in a Riemannian setting, it is easier to show some parallel condition than some symplectic ones, etc. Here, I try to find a condition on $\varphi$ without applying any restriction on it; moreover, I am almost sure I have already read such a condition!

Answer 1

I finally found the conditions for $\varphi|_H$ to be integrable: first, one needs to have a compatibility condition relying $\eta$ and $\varphi$: \begin{align} \forall X,Y \in \Gamma(H),~ \mathrm{d}\eta(\varphi X, Y) &= - \mathrm{d}\eta(X,\varphi Y), & \mathrm{d}\eta(\varphi X,\varphi Y) = \mathrm{d}\eta(X,Y), \end{align} and secondly, the Nijenhuis tensor $N_{\varphi}$, restribted to $H$, has to take values in $\mathbb{R}\xi$. Here is a proof:

Notice that $\varphi\left(\varphi^2+1\right) = 0$, and hence (the $\mathbb{C}$-linear extension of) $\varphi$ is diagonalizable in $TM\otimes \mathbb{C}$, with $$ TM\otimes \mathbb{C} = \mathbb{C}\xi \oplus H^{1,0}\otimes H^{0,1}, $$ with $H^{1,0} = \left\{X - i\varphi X \mid X \in H \right\}$ and $H^{0,1} = \overline{H^{1,0}} = \left\{X+i\varphi X \mid X \in H \right\}$.

For $V$ a section of $TM\otimes \mathbb{C}$, we have the decomposition $$ V = \eta(V)\xi + V^{1,0} + V^{0,1}. $$ A direct computation shows that $V+ i \varphi V = \eta(V)\xi + 2 V^{0,1}$. It follows that $$V \in H^{1,0} \iff \eta(V) = 0 \text{ and } V^{0,1} = 0 \iff V + i \varphi V = 0.$$ Now, if $X$ and $Y$ are in $H$, a direct computations shows that, writing $V = [X-i\varphi X,Y-i\varphi Y]$, $$ V = [X,Y] - [\varphi X,\varphi Y] -i [\varphi X,Y] -i [X,\varphi Y] $$ Hence, using the fact that $\eta\circ \xi=0,~\varphi \xi = 0,~ \varphi ^2 = -\mathrm{id} + \eta\otimes \xi$, and that $\eta([A,B]) = -\mathrm{d}\eta(A,B)$ for $A,B \in H$:
\begin{align} V + i\varphi V &= \left(-\varphi^2[X,Y] + \varphi[\varphi X,Y] + \varphi[X,\varphi Y] - [\varphi X,\varphi Y] - \mathrm{d}\eta(\varphi X,\varphi Y)\otimes \xi\right) \\ &+ i\varphi \left(-\varphi^2[X,Y]+ \varphi[\varphi X,Y] + \varphi[X,\varphi Y] - [\varphi X,\varphi Y] \right) \\ &+ \left[\left(\mathrm{d}\eta(\varphi X,\varphi Y) - \mathrm{d}\eta(X,Y) \right) - i\left(\mathrm{d}\eta(\varphi X,Y) + \mathrm{d}\eta(X,\varphi Y) \right)\right]\otimes \xi \\ &= \left(N_{\varphi}(X,Y) - \mathrm{d}\eta(\varphi X,\varphi Y)\otimes \xi \right) + i\varphi N_{\varphi}(X,Y) \\ &+ \left[\left(\mathrm{d}\eta(\varphi X,\varphi Y) - \mathrm{d}\eta(X,Y) \right) - i\left(\mathrm{d}\eta(\varphi X,Y) + \mathrm{d}\eta(X,\varphi Y) \right)\right]\otimes \xi. \end{align} This equation yields the following: \begin{align} V \in H^{1,0} & \iff \left\{\begin{array}{rcl}N_{\varphi}(X,Y) - \mathrm{d}\eta(\varphi X,\varphi Y)\otimes \xi &=& 0 \\ \varphi N_{\varphi}(X,Y) &=& 0 \\ \mathrm{d}\eta(X,Y) - \mathrm{d}\eta(\varphi X , \varphi Y) &=& 0 \\ \mathrm{d}\eta(\varphi X, Y) + \mathrm{d}\eta(X,\varphi Y) &=& 0 \end{array}\right. \\ &\iff \left\{\begin{array}{rcl} N_{\varphi}(X,Y) &=& \mathrm{d}\eta(\varphi X,\varphi Y) \otimes \xi \\ N_{\varphi}(X,Y) & \in & \mathbb{R}\xi \\ \mathrm{d}\eta(\varphi X, \varphi Y) &=& \mathrm{d}\eta(X,Y) \\ \mathrm{d}\eta(\varphi X, Y) &=& -\mathrm{d}\eta(X,\varphi Y) \end{array} \right. . \end{align} One can easily show, thanks to the definition of $N_{\varphi}$ and the fact that $\eta\circ \varphi = 0$, that under the third condition ($\mathrm{d}\eta(\varphi X,\varphi Y) = \mathrm{d}\eta(X,Y)$), the two first conditions are equivalent. Finally: $$ [H^{1,0},H^{1,0}] \subset H^{1,0} \iff\forall X,Y \in \Gamma(H),~\left\{\begin{array}{rcl} \mathrm{d}\eta(X,Y) &=& \mathrm{d}\eta(\varphi X,\varphi Y) \\ \mathrm{d}\eta(\varphi X,Y) &=&-\mathrm{d}\eta(X,\varphi Y) \\ N_{\varphi}(X,Y) &\in& \mathbb{R}\xi\end{array} \right. . $$ One can also remark that the two first conditions are equivalent for vector fields tangent to $H$. This yields this final equivalence: $$ \varphi|_H \text{ is integrable } \iff \left\{\begin{array}{l} \mathrm{d}\eta \text{ is of type } (1,1) \text{ with respect to } \varphi \\ N_{\varphi} \text{ restricted to } \Gamma(H) \text{ takes values in } \mathbb{R}\xi \end{array}\right. $$