Variable-radius injectivity for normal addition

By [L1], for every $y\in S$ there is some ${\epsilon }_y>0$ such that the normal addition map is a diffeomorphism on $V_{{\epsilon }_y}(y):=\{(y^{\mathrm{\prime }},v^{\mathrm{\prime }})\in N^{\perp }S:\mathrm{\parallel }{y}^{\mathrm{\prime }}-y\mathrm{\parallel }<{\epsilon }_y,\mathrm{\parallel }{v}^{\mathrm{\prime }}\mathrm{\parallel }<{\epsilon }_y\}\mathrm{.}$ Define $r(y):=\min (1,\sup \{\epsilon >0:E{\mathrm{\mid }}_{V_{\epsilon }(y)}\text{ is a diffeomorphism}\})\mathrm{.}$ Then $r(y)>0$ for every $y\in S$.

The function $r$ is continuous. Indeed, if $\mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel }<r(y)$ and $0<\epsilon <r(y)-\mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel }$, then $V_{\epsilon }(y^{\mathrm{\prime }})\subseteq V_{r(y)}(y)$, so $E{\mathrm{\mid }}_{V_{\epsilon }(y^{\mathrm{\prime }})}$ is also a diffeomorphism. Hence $r(y^{\mathrm{\prime }})\ge r(y)-\mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel }$. Swapping $y$ and $y^{\mathrm{\prime }}$ gives $\mathrm{\mid }r(y)-r(y^{\mathrm{\prime }})\mathrm{\mid }\le \mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel }$.

By [L2], choose a smooth positive function $\delta :S\to (0,\mathrm{\infty })$ with $\delta (y)\le r(y)\mathrm{/}2$ for every $y\in S$.

Suppose $E(y,v)=E(y^{\mathrm{\prime }},v^{\mathrm{\prime }})$ with both points in ${\mathrm{\Omega }}_{\delta }$, and assume without loss of generality that $r(y^{\mathrm{\prime }})\le r(y)$. Then $\mathrm{\parallel }y-y^{\mathrm{\prime }}\mathrm{\parallel }=\mathrm{\parallel }v-v^{\mathrm{\prime }}\mathrm{\parallel }\le \mathrm{\parallel }v\mathrm{\parallel }+\mathrm{\parallel }{v}^{\mathrm{\prime }}\mathrm{\parallel }<\delta (y)+\delta (y^{\mathrm{\prime }})\le \frac{r(y)}{2}+\frac{r(y^{\mathrm{\prime }})}{2}\le r(y)\mathrm{.}$ Also $\mathrm{\parallel }v\mathrm{\parallel }<\delta (y)\le r(y)\mathrm{/}2<r(y)$ and $\mathrm{\parallel }{v}^{\mathrm{\prime }}\mathrm{\parallel }<\delta (y^{\mathrm{\prime }})\le r(y^{\mathrm{\prime }})\mathrm{/}2\le r(y)\mathrm{/}2<r(y)$. Therefore both $(y,v)$ and $(y^{\mathrm{\prime }},v^{\mathrm{\prime }})$ lie in $V_{r(y)}(y)$, where $E$ is injective by step 1.1. Hence $(y,v)=(y^{\mathrm{\prime }},v^{\mathrm{\prime }})$. So $E$ is injective on ${\mathrm{\Omega }}_{\delta }$.