Real and complex k-planes in convex hypersurfaces

Abstract

It is shown that that the rank of the second fundamental form (resp. the Levi form) of a ${\mathcal{C}}^2$-smooth convex hypersurface $M$ in ${\mathbb{R}}^{n+1}$ (resp. ${\mathbb{C}}^{n+1}$) does not exceed an integer constant $k<n$ near a point $p\in M,$ then through any point $q\in M$ near $p$ there exists a real (resp. complex) $(n-k)$-dimensional plane that locally lies on $M.$