= Disk bundle
{title2=$D(E)$}
For a real <vector bundle> $E\to B$ with a <fiber metric>, the <disk bundle> is
$$
D(E)=\{v\in E:\|v\|\leq1\}.
$$
Its fiberwise boundary is the <sphere bundle> $S(E)=\{\|v\|=1\}$. The radial <homotopy> $H_t(v)=(1-t)v$ gives a <deformation retraction> onto the <zero section>, so $D(E)$ is <homotopy equivalent> to $B$.
If $B$ is a <smooth manifold> without boundary and the rank is positive, the total-space boundary is exactly $S(E)$. When $B$ is a <manifold with boundary>, the base boundary also contributes: the boundary faces are $S(E)$ and $D(E)|_{\partial B}$, meeting along $S(E)|_{\partial B}$. For the <trivial vector bundle> of rank one over $[0,1]$, the <disk bundle> is the rectangle $[0,1]\times[-1,1]$. Its top and bottom sides form the <sphere bundle>; its left and right sides come from the base boundary. The intersections are corners.
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