Submanifolds of the Grassmann manifold with vanishing Chern forms (Q1070085)

Motivated by \textit{P. Griffiths}' paper in Duke Math. J. 45, 427-512 (1978; Zbl 0409.53048) the author gives conditions on the Chern forms \(c_ k\) of the universal bundle E on the Grassmannian G(n,N) which imply that certain submanifolds have a parallel subbundle in E. For example, if \(M^ r\) is a submanifold, \(r\geq 2\), \(c_ k|_ M\neq 0\) and \(c_{k+1}|_ M=0\) for some \(k\leq (r+N-n-2)/(N-n),\) then \(E|_ M\) has a constant subbundle of rank n-k.