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The disk bundle of a metric vector bundle consists of the vectors of norm at most one. Its boundary is the sphere bundle $S(E)$, and radial contraction makes $D(E)$ homotopy equivalent to its zero section.

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 $\Vert v\Vert =1$.