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On sectioning multiples of the nontrivial line bundle over Grassmannians

Horanská, Ľubomíra — 1998

Proceedings of the 17th Winter School "Geometry and Physics"

Let $G_{n, k}$ ( ${\tilde{G}}_{n, k}$ ) denote the Grassmann manifold of linear $k$ -spaces (resp. oriented $k$ -spaces) in $ℝ^{n}$ , $d_{n, k} = k (n - k) = dim G_{n, k}$ and suppose $n \geq 2 k$ . As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n, k} + 1)$ -fold Whitney sum $(d_{n, k} + 1) ξ_{n, k}$ of the nontrivial line bundle $ξ_{n, k}$ 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 $s ξ_{n, k}$ 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...

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