Disk, sphere, and Thom spaces of a metric vector bundle

Let $\xi =(E\stackrel{\pi }{\to }B,h)$ be a rank-$n$ real vector bundle equipped with a continuous fiber metric. Define $D_h(\xi )=\{v\in E:\mathrm{\parallel }v{\mathrm{\parallel }}_h\le 1\},S_h(\xi )=\{v\in E:\mathrm{\parallel }v{\mathrm{\parallel }}_h=1\},$ and define the Thom space by the based quotient ${\mathrm{Th}}_h(\xi )=D_h(\xi )\mathrm{/}{S}_h(\xi )\mathrm{.}$ Here $X\mathrm{/}\mathrm{\varnothing }$ means $X_{+}$, so the rank-zero convention is already included.

For two supplied metrics $h$ and $k$, the canonical radial homeomorphism of pairs is $r_{h,k}(0_b)=0_b,$ $r_{h,k}(v)=\frac{\mathrm{\parallel }v{\mathrm{\parallel }}_h}{\mathrm{\parallel }v{\mathrm{\parallel }}_k}v(v\mathrm{\ne }0)\mathrm{.}$ It preserves the base and normalized radius, has inverse $r_{k,h}$, and descends to the quotient. Continuity at the zero section follows in a bundle chart because the ratio of the two norms on the unit sphere is locally bounded above and below. The positive-definite interpolation $h_t=(1-t)h+tk$ gives the canonical radial isotopy $r_{h,h_t}$.

This definition is choice-free once the metric is supplied. For the empty base all three spaces are empty except for the quotient basepoint; for rank zero, $D_h(\xi )=B$, $S_h(\xi )=\mathrm{\varnothing }$, and ${\mathrm{Th}}_h(\xi )=B_{+}$. The zero vector is fixed, sphere and disk endpoints are preserved, and the formulas for $h=k$ are the identity.