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The rank of vector fields on Grassmannian manifolds. (English) Zbl 0656.57012

Geometry and physics, Proc. Winter Sch., Srní/Czech. 1987, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 16, 113-117 (1987).
[For the entire collection see Zbl 0634.00015.]
The tangent bundle of the real Grassmann manifold \(G_{m,p}\) of p-planes in \({\mathbb{R}}^ m\) is known to have the form \(TG_{m,p}\cong \text{Hom}(\gamma,\gamma^{\perp}),\) where \(\gamma\) is the canonical p- plane bundle. There is a nowhere vanishing tangential vectofield on \(G_{m,p}\) \((=\) section of \(\text{Hom}(\gamma,\gamma^{\perp})\) of rank \(\geq 1\) everywhere) iff \(m\) is even and \(p\) is odd. Using singularity techniques, the authors show for \(p=2r+1:\) If there is a section of rank \(>1\) everywhere, then all of \(\binom{(m/2)-1}r\), \(\binom{m/2}r\), and \(\binom{m/2}{r+1}\) are even.
Reviewer: M.Raußen

MSC:

57R25 Vector fields, frame fields in differential topology

Citations:

Zbl 0634.00015
Full Text: EuDML