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Hello,

$CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point.

I can't see what the happening in Non-Levi flat points. Sorry for vague question and if it is trivial.. .... but basically i want to understand the non levi flat point such that i can easily determine all non levi flat point of given CR manifold...

Any comment and suggestion are welcome.. Thanks in advance.

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    $\begingroup$ The boundary of a strongly pseudonvex domain . $\endgroup$
    – Mohan Ramachandran
    Commented Feb 18, 2012 at 2:39
  • $\begingroup$ @ramachandran sir, thanks for the comment... Ohkk probably that is main point to use the terminology "not flat" cause non levi flat point lies in boundary of some pseudconvex domain... Actually i am first time coming across with this subject so it will take some time for assimilation.. Thanks for helping.. $\endgroup$
    – zapkm
    Commented Feb 18, 2012 at 6:58

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Probably the most easy example to understand is the round sphere in $\mathbb{C}^2$. It is strictly convex, so that in particular it is strictly pseudo-convex, and very not Levy-flat.

For a submanifold $M$, Levy flatness is equivalent to the integrability of $TM\cap i TM$. If you compute this plane distribution in the case of the unit sphere, you will see that it is the orthogonal to the Hopf fibration, and in particular as far from being integrable as possible.

This is short, but I hope there are sufficient key words so that you can study the subject more deeply in books.

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  • $\begingroup$ @benoit kloeckner.. Thanks for the help... certainly i wanted this much only... $\endgroup$
    – zapkm
    Commented Feb 18, 2012 at 7:00
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    $\begingroup$ One more keyword: contact manifolds. $\endgroup$
    – Elizabeth S. Q. Goodman
    Commented Feb 18, 2012 at 8:33

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