## 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

Zbl 0634.00015
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