I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question: In the proof of Lemma 7.4, He let $d$ be the signed distance function of $B_{\delta}(y(s))\cap y(S)$ for $p=y(s)$ and let $f=d+\frac{\epsilon}{2}d^2$, where $y:S\to M$ is an $\epsilon$-convex immersion and $\delta$ such that $y|_{y^{-1}(B_{\delta}(y(s)))}$ is an embedding. Then he concluded that $f$ is smooth on $\{|d|\leq r_1\}$, where $r_1$ is the focal distance of the whole hypersurface $y(S)$. I can't understand why the locally distance function $d$ is smooth in this domain.
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Choose a unit normal field $n$ to the surface $S$. Consider the map $m\colon S\times \mathbb{R}\to M$ defined by $m\colon(p,t)\mapsto \exp_p(t\cdot n_p)$.
Note $m$ is smooth and locally injective in $S\times (-r_1,r_1)$. Then argue as in the Gauss lemma.
answered Oct 4, 2022 at 12:38
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$\begingroup$ I think this can be understand as the index theorem for focal points? $\endgroup$– eulershiCommented Oct 4, 2022 at 12:42
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$\begingroup$ @eulershi yes. $ $ $ $ $ $ $\endgroup$ Commented Oct 4, 2022 at 12:43
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$\begingroup$ But I'm still confused that the $\delta$ in locally injective domain can be made uniformly for all $s\in S$, though $S$ is compact. $\endgroup$– eulershiCommented Oct 4, 2022 at 13:06
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2$\begingroup$ @eulershi you can do it locally and cover the surface by finite number of good neighborhoods. $\endgroup$ Commented Oct 4, 2022 at 22:27