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,* 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 $\gamma_{k,N}$.