Let $G_{n,k}$ (${\tilde{G}}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${ℝ}^n$, $d_{n,k}=k(n-k)=\text{dim}{G}_{n,k}$ and suppose $n\ge 2k$. 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, $s_{n,k}\le d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1}=d_{n,1}+1$. Using results of J. Korbaš and P. Sankaran [Proc. Indian Acad. Sci., Math. Sci. 101, No. 2, 111-120 (1991; Zbl 0745.55003)], S. Gitler and D. Handel [Topology 7, 39-46 (1968; Zbl 0166.19405)] and the Dai-Lam level of ${\tilde{G}}_{n,k}$ with !

EUDML-ID : urn:eudml:doc:221449
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@article{701615,
     title = {On sectioning multiples of the nontrivial line bundle over Grassmannians},
     booktitle = {Proceedings of the 17th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1998},
     pages = {[59]-64},
     mrnumber = {MR1662726},
     zbl = {0930.55008},
     url = {http://dml.mathdoc.fr/item/701615}
}

Horanská, Ľubomíra. On sectioning multiples of the nontrivial line bundle over Grassmannians, dans Proceedings of the 17th Winter School "Geometry and Physics", GDML_Books,  (1998), pp. [59]-64. http://gdmltest.u-ga.fr/item/701615/