Let $G\_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle 
$$
\gamma\_{k,N}: \mathbb{R}^k\longrightarrow E(\gamma\_{k,N})\longrightarrow G\_k(\mathbb{R}^N)
$$
where 
\begin{eqnarray\*}
E(\gamma\_{k,N})&=&\{(V,v)\mid V\in G\_k(\mathbb{R}^N), v\in V\}
\\&=&V\_k(\mathbb{R}^N)\times\_{GL(k)}\mathbb{R}^k.
\end{eqnarray\*}
Here $V\_k(\mathbb{R}^N)$ is the Stiefel manifold consisting of $k$-tuples or linearly independent vectors in $\mathbb{R}^N$.
In particular, when $k=1$, we have the canonical Hopf line bundle
$$
\gamma\_{1,N}: \mathbb{R}\longrightarrow S^N\times\_{ \mathbb{Z}/2} \mathbb{R}\longrightarrow \mathbb{R}P^N.
$$


**Question.** Are there references/results for the order (the order of a vector bundle is the smallest positive integer $n$ such that the $n$-fold self-Whitney sum of the vector bundle is trivial) of $\gamma\_{k,N}$?


In the paper [*Vector fields on spheres,*](http://www.maths.ed.ac.uk/~aar/papers/adams2.pdf) J.F. Adams, 1962, Theorem 7.4, it is proved 
$$
\text{order}(\gamma\_{1,N})=2^{\phi(N)}
$$
where $\phi(N)$ is the number of positive integers no larger than $N$ and congruent to $0,1,2,4$ mod $8$. I want to find references/generalizations to $\gamma\_{k,N}$.



self-Whitney sum of the canonical vector bundle on Grassmannians
Let me expand the comment above a bit. Consider the tautological bundle $\tau\to\mathbb C P^2$. It is complex and has total Chern class $c(\tau)=1+a$, where $a\in H^2(\mathbb C P^2)$ generates the cohomology ring of $\mathbb C P^2$. Viewed as a real bundle, it has $$p\_1(\tau\_{\mathbb R})=-c\_2(\tau\oplus\bar\tau)=-c\_1(\tau)c\_1(\bar\tau)=a^2$$ because $\tau\_{\mathbb R}\otimes\_{\mathbb R}\mathbb C\cong \tau\oplus\bar\tau$. But $a^2$ generates $H^4(\mathbb C P^2)\cong\mathbb Z$, so $\tau\_{\mathbb R}$ has order $\infty$.


So, as soon as $N\_0$ is large enough such that $G\_k(\mathbb R^{N\_0})$ classifies $\tau\_{\mathbb R}$,
you get order $\infty$ for $\gamma\_{2,N\_0}$. By adding a trivial bundle to $\tau$, the same argument shows that the order of $\gamma\_{2+n,N\_0+n}$ is $\infty$ as well.


