On sectioning multiples of the nontrivial line bundle over Grassmannians

Abstract

summary:Let $G_{n,k}$ ($\widetilde {G}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${\Bbb R}^{n}$, $d_{n,k} = k(n-k) = \text {dim} G_{n,k}$ and suppose $n \geq 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n,k}+1)$-fold Whitney sum $(d_{n,k}+1)\xi_{n,k}$ of the nontrivial line bundle $\xi_{n,k}$ over $G_{n,k}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($= s_{n,k}$) such that the vector bundle $s\xi_{n,k}$ admits a nowhere vanishing section ? Obviously, $s_{n,k} \leq d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1} = d_{n,1}+1$. Using results of {\it J. Korba\v{s}} and {\it P. Sankaran} [Proc. Indian Acad. Sci., Math. Sci. 101, No. 2, 111-120 (1991; Zbl 0745.55003)], {\it S. Gitler} and {\it D. Handel} [Topology 7, 39-46 (1968; Zbl 0166.19405)] and the Dai-Lam level of $\widetilde {G}_{n,k}$ with