Real and complex -planes in convex hypersurfaces

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