In the book "Cohomology Operations" by Steenrod and Epstein, at page 56, the author says that the existence of a field of $k$-frames (i.e. a function that maps a point of the sphere to a $k$-frame tangent to the point) is equivalent to the existence of a section to the fibre bundle $$V_{k+1}(\mathbb{R}^n)\longrightarrow V_1(\mathbb{R}^n)=S^{n-1}$$ I do not have that much familiarity with fibre bundles, and I would like to know how they are equivalent.
Thanks to everybody who will answer.
PS: $V_k(\mathbb{R}^n)$ is the Stiefel manifold of $k$-frames in $\mathbb{R}^n$. A $k$-frame is a set of $k$ orthonormal vectors.
Think of $S^{n-1}$ as the unit sphere in $\mathbb{R}^n$. Then, we can identify the tangent space $T_x S^{n-1}$ at $x \in S^{n-1}$ with the hyperplane in $\mathbb{R}^n$ that is perpendicular to $x$ regarded as a vector, and furthermore this identification is continuous in $x$.
Now, given a field of $k$-frames $\{\eta_1(x), \ldots, \eta_k(x)\}$, the map $\sigma: x \mapsto (x, \eta_1(x), \ldots, \eta_k(x))$ is a section of $V_{k+1}(\mathbb{R}^n) \to S^{n-1}$. Conversely, given a section $\sigma: S^{n-1} \to V_{k+1}(\mathbb{R}^n)$, we can write $\sigma(x) = (x, \eta_1(x), \ldots, \eta_k(x))$, and $\{\eta_1(x), \ldots, \eta_k(x)\}$ is a $k$-frame field on $S^{n-1}$.