Sphere bundle ($S(E)$)

After choosing a fiber metric on a rank-$d$ vector bundle $E\to B$, its sphere bundle consists of the unit vectors and has fiber $S^{d-1}$. Different choices of metric give isomorphic sphere bundles.

The unit tangent bundle of a Riemannian manifold $M$ is the sphere bundle of its tangent bundle. Its points are pairs $(x,v)$ with $x\in M$, $v\in T_{x}M$, and $\Vert v\Vert =1$.

The real Stiefel manifold $V_k({\mathbb{R}}^n)$ consists of ordered orthonormal $k$-frames in ${\mathbb{R}}^n$. In particular, $V_2({\mathbb{R}}^{m+1})$ is naturally the unit tangent bundle of $S^m$.

The Euler class of $T{S}^{2n}$ evaluates to $2$. Its Gysin sequence of a sphere bundle givesAll products of positive-degree classes vanish.