The Wikipedia article on (real) [Grassmannians](http://en.wikipedia.org/wiki/Grassmannian#Schubert_cells) gives a simple argument that the Euler characteristic satisfies a recurrence relation $$\chi G\_{n,r} = \chi G\_{n-1,r-1} + (-1)^r \chi G\_{n-1,r}$$. This implies that the euler characteristic is zero if and only if $n$ even and $r$ odd. So $\chi G\_{6,3} = 0$, for example. Basic obstruction theory on manifolds tells us that if $\chi M=0$ then there is a non-vanishing vector field on $M$. 


  



Does anyone have any simple, explicit examples of such vector fields? Let's say $G\_{n,1}$ for $n$ even does not count. Not to say those aren't interesting examples -- I'd love solutions that simple. But it's not immediately clear how they can be generalized. 



Formulas for vector fields on Grassmannians?
Identify $\mathbb R^2$ with $\mathbb C$ and consider the $S^1$ action on $\mathbb R^{2n} \simeq \mathbb C^n$ induced by cordinatewise complex multiplication. These of course lead to the trivial examples on $G\_{2n,1}$. For $n$ even and $r$ odd the very same examples do the trick. One has just to observe that these $S^1$ actions have no invariant odd dimensional subspaces, and therefore induce $S^1$-actions without fixed points on $G\_{n,r}$.


