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Let $M$ be a smooth manifold and let $TM\otimes\mathbb C$ be its complexified tangent bundle. Let $H$ be a complex subbundle of $TM\otimes\mathbb C$.

It can be happened that $H$ and $\bar H$ are involutive (i.e $[H,H]\subset H$ and $[\bar H,\bar H]\subset \bar H$) but $H\oplus \bar H$ is not involutive.

Can you please provide me in examples of this situation?

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  • $\begingroup$ Just out of curiosity, is there a reason you started with $TM\otimes\Bbb C$ rather than an arbitrary complex vector bundle on $M$? $\endgroup$ – Ted Shifrin Nov 30 '17 at 22:23
  • $\begingroup$ Yes, because I am looking for a non Levi flat CR manifold. Actually I am constructing one .. $\endgroup$ – Ronald Nov 30 '17 at 22:29
  • $\begingroup$ Aha. OK. I still don't have an answer yet. So you're doing the special case where $H$ is the maximal complex subbundle of the real tangent bundle, which is far more specific than the question you posed. $\endgroup$ – Ted Shifrin Nov 30 '17 at 22:35
  • $\begingroup$ But the example maybe is easy! Do you think $\{(z,w)\in\mathbb C^2; z+\bar z=w\bar w\}$ would work for my question? $\endgroup$ – Ronald Nov 30 '17 at 22:54

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