Let $M \subset \mathbb C^n$ be a $C^\infty$ embedded real hypersurface, where $n\ge 1$, and let $J:T\mathbb C^n\to T\mathbb C^n$ denote the standard complex structure. For each $p\in M$, define the real CR tangent space

\begin{align*} H_p(M):=T_pM\cap J(T_pM). \end{align*}

If $\rho:V\to\mathbb R$ is a local defining function on an [open set](/page/Open%20Set) $V\subset\mathbb C^n$, meaning $M\cap V=\rho^{-1}(\{0\})$ and $d\rho_q\ne0$ for every $q\in M\cap V$, define the real one-form $\theta_\rho$ on $M\cap V$ by

\begin{align*} (\theta_\rho)_q(v):=d\rho_q(Jv) \quad \text{for } q\in M\cap V \text{ and } v\in T_qM. \end{align*}

Let $\Gamma(E)$ denote the space of smooth local sections of a smooth vector bundle $E$. Say that $M$ is Levi-flat if, for every local defining function $\rho$, one has

\begin{align*} d\theta_\rho(X,Y)=0 \end{align*}

for all smooth local real sections $X,Y\in\Gamma(H(M))$ on the domain where $\theta_\rho$ is defined. Then $M$ is Levi-flat if and only if, for every $p\in M$, there is a neighbourhood $U\subset M$ of $p$ such that the smooth distribution $H(M)|_U$ is tangent to a smooth local foliation of $U$ whose leaves are immersed complex hypersurfaces of $\mathbb C^n$. For $n=1$, this is understood as the zero-dimensional foliation by points.