I have the following question on vector bundles which I am struggling to understand:
Consider the tangent bundle $TS^2$ with an inner product on the fibres, and let $Y\subset TS^2$ consist of all the tangent vectors of unit length. Show that $TS^2$ induces on $Y$ the structure of a principal $S^1$ bundle over $S^2$.
I am struggling to understand the last part of the question. Is this regarding $TS^2$ over $Y$ as a vector bundle, which is to have the same structure group as "the structure of $S^1$ bundle over $S^2$"? Though I am not certain as to what it. I thought it might be a vector bundle or principal bundle, but $S^1$ has a lower dimension than $S^2$ so being a bundle 'over' $S^2$ doesn't quite make sense to me.