what does following matrix says geometrically Let $M\subset \mathbb C^2$ be a hypersurface defined by $F(z,w)=0$. Then for some point $p\in M$, I've
$$\text{ rank of }\left(
    \begin{array}{ccc}
     0 &\frac{\partial F}{\partial z} &\frac{\partial F}{\partial w} \\
 \frac{\partial F}{\partial z} &\frac{\partial^2 F}{\partial ^ 2z} &\frac{\partial^2 F}{\partial z\partial w} \\
 \frac{\partial F}{\partial w} &\frac{\partial^2 F}{\partial w\partial z} &  \frac{\partial^2 F}{\partial w^2} \\
                    \end{array}
                  \right)_{\text{ at p}}=2.$$
What does it mean geometrically? Can anyone give a geometric picture near $p$? 
Any comment, suggestion, please.
Edit: Actually I was reading about Levi flat points and Pseudo-convex domains.  I want to understand the relation between these two concepts. A point p for which the rank of the above matrix is 2 is called Levi flat. If the surface is everywhere Levi flat then it is locally equivalent to $(0,1)\times \mathbb{C}^n$, so I have many examples....but what will happen for others for example take the three sphere in $\mathbb{C}^2$ given by $F(z,w)=|z|^2+|w|^2−1=0$.  This doesn't satisfy the rank 2 condition.  Can I have precisely these two situations?
 A: Let $p=(z_0,w_0)$ and define
$G(z,w)=F(z,w)-(z_0,w_0)$.
Then the matrix is
$$
\left(
\begin{matrix}
  G   &  G_z    &  G_w      \cr
  G_z & (G_z)_z & (G_z)_w \cr
  G_w & (G_w)_z & (G_w)_w \cr
\end{matrix}
\right)_{\text{at }p}
$$
Since $G(p)=0$.
Is that any help?
A: Here is a partial answer: I will givea geometric interpretation of Levi flatness/pseudoconvexity.  To fix some notation, let $j$ be endomorphism of the tangent bundle to $\mathbb{C}^2$ induced by its complex structure.  (I'm being a bit pedantic, normally we say it is the complex structure, but I want to make it very clear what I am describing.)
If you have a real hypersurface $\Sigma$ in $\mathbb{C}^2$, its tangent bundle has a preferred complex line bundle inside of it.  This consists of those vectors in $TM$ such that $j v$ is also in $TM$. Let $\xi$ be this subbundle.  We say that $\xi$ is Levi-flat if this distribution is (locally) integrable in the sense of Frobenius.
So what does this mean, geometrically?  Suppose that $\Sigma$ is Levi-flat in an open neighbourhood of $p \in \Sigma$.  Then, by the Frobenius integrability theorem, you can find a local function $G \colon \Sigma \to \mathbb{R}$ whose level sets have $j$ invariant tangent spaces, i.e. the level set is a complex (local) submanifold of $\mathbb{C}^2$.  Again, since we are working locally, this allows you to describe the neighbourhood of $p$ as being of the form $(-\epsilon, \epsilon) \times D^2(\epsilon)$, where $D^2$ is the disk in $\mathbb{C}$.
Levi convexity is a bit harder to explain without appealing to the Levi form.  See the reference I gave in the comments above for some definitions and discussion of the concept.  In particular, a convex hypersurface in $\mathbb{C}^2$ is Levi convex.  
The key fact about flatness/convexity has to do with holomorphic disks whose boundaries are in $\Sigma$.  If $\Sigma$ is flat, you can foliate $\Sigma$ locally by such disks.  If $\Sigma$ is strictly pseudoconvex, then only the boundary of the disk touches $\Sigma$, the interior of the disk is forced to lie in the interior region bounded by $\Sigma$.  (For instance, think of the unit sphere $S^3$ as the typical example of a pseudoconvex hypersurface.  Any holomorphic disk with boundary in $S^3$ lives inside the unit ball -- furthermore, only its boundary is allowed to touch the $S^3$.)
In an example like the one you gave, the complex line is $\ker dF \cap \ker dF \circ j$.  You then want to compute the two form $\omega := -d (dF \circ j)$ on a pair of (nonzero) vectors $v, jv$, $v \in \xi$.  If this is positive, then it is pseudoconvex (at this point).  If it is zero, it is Levi-flat.
