CR-structures and Riemannian geometry of hypersurfaces of \({\mathbb{C}}^ n\) (Q1073982)

The author studies some relations between the Riemannian structure and the Cauchy-Riemann structure of a hypersurface M in \({\mathbb{C}}^ n\). Among the results obtained there are: M is strictly Levi-convex if the curvature of each complex section is positive. If M is Levi-flat the Ricci curvature is non positive iff M is a Riemannian manifold with identically zero sectional curvature. The Levi flat surfaces with identically zero sectional curvature are isomorphic (as CR-manifolds) to an open subset of \({\mathbb{C}}^{n-1}\times {\mathbb{R}}\) or of \({\mathbb{C}}^{n-1}\times S^ 1(r)\) \((:=\{(z_ 1,...,z_ n)\in {\mathbb{C}}^ n| \quad | z_ n| =r\}).\)