Levi-flat hypersurface

Remarks

The condition does not depend on the choice of $\rho$ or of the coordinates. Example: $M = \set{\operatorname{Im} z_1 = 0} \subset \C^n$ with $\rho = \operatorname{Im} z_1 = (z_1 - \bar z_1)/(2i)$. All mixed derivatives $\partial^2 \rho / \partial z_j \partial \bar z_k$ vanish, so the Levi form is $0$ and $M$ is Levi-flat. Non-example: the unit sphere $\set{\abs{z_1}^2 + \cdots + \abs{z_n}^2 = 1}$ with $\rho = \abs{z}^2 - 1$. Its Levi form is $\abs{w_1}^2 + \cdots + \abs{w_n}^2$, which is not zero on $H_p$, since $H_p \ne 0$ when $n \ge 2$. For a smooth hypersurface, an equivalent condition is that the complex tangent distribution $p \mapsto H_p$ is integrable. Then $M$ is locally foliated by complex hypersurfaces of $X$; this foliation is called the Levi foliation. A Levi-flat hypersurface is a CR manifold of hypersurface type.