# every where levi flat

"Suppose $N$ is $2n-1; n\geq 2$ dimensional $CR$ manifold and everywhere Levi flat, then it will be locally $CR$ equivalent to $S^1\times \mathbb C^{n-1}.$"

Above statement can be found in Loop space as complex manifolds paper by Laszlo Lempert, J. Differential geometry 38(93),519-543.http://intlpress.com/JDG/archive/1993/38-3-519.pdf. But I am not able to see it. May be i didn't get the meaning of Levi flat properly..

Can someone please provide the reference material to understand much about Levi flat and Non levi flat point manifolds.

Where the things are wrong in the following example:

Take $N= (0,1)\times \mathbb C\equiv (0,1)\times \mathbb R^2.$ Let $t, x, y$ be coordinate. For $p\in N$, $T_pN= \mathbb R \{\frac{\partial}{\partial t},\frac{\partial}{\partial x},\frac{\partial}{\partial y}\}$. Define $H_p^{1,0}N= \{a.\frac{\partial}{\partial z}: a\in \mathbb C\}$. Then $H_p^{1,0}N$ defines $CR$ structure on $N$. And with this structure, $N$ is every where Levi flat.

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@Pradip: Nothing is wrong with your example. It is indeed Levi-flat and it is locally CR-equivalent to $S^1\times\mathbb{C}^1$. Howard Jacobowitz and Al Bogess have nice little books on CR-geometry (I forget their exact title), and these would be a good introduction, if that is what you want. – Robert Bryant Jan 9 '12 at 4:30
@Robert Bryant sir, Thanks for the comment.. I got the book by Howard Jacobowitz.. title is AN INTRODUCTION TO CR STRUCTURES. It is an AMS monograph... – zapkm Jan 9 '12 at 12:59