Vector fields on real flag manifolds (Q1067263)

This paper determines which of the flag manifolds \(F=F(n_ 1,...,n_ r)\) are parallelizable. This is the space of mutually orthogonal r-tuples of subspaces of dimensions \(n_ 1,...,n_ r\) in \({\mathbb{R}}^ n\), \(n=n_ 1+...+n_ r\). It is shown that the parallelizable F's are precisely those given by \(F(1,3)={\mathbb{R}}P^ 3\), \(F(1,7)={\mathbb{R}}P^ 7\), and the classical flag manifolds F(1,1,...,1). Further, the author says a few words about the span of F, the number of linearly independent vector fields. In particular, \(span(F(1,1,2))=3.\) (Note: See \textit{P. Sankaran} and \textit{P. Zvengrowski} [Pac. J. Math. 122, 455-458 (1986; Zbl 0557.14030)] for recent additional work.)