Referring to G. Perelman, Proof of the soul conjecture by Cheeger and Gromoll. Given a distance-nonincreasing retraction $P$ from an open complete manifold of nonnegative curvature onto its soul $S$, one wants to prove
$$P(\exp_xtν)=x \text{ for every } x\in S \text{ and } t≥0. \tag{$A$}$$
$(B)$ For any geodesic $\gamma \subset S$ and any unit vector field $\nu$ in its normal bundle which is parallel along $\gamma$, the "horizontal" curves $\gamma_t, \gamma_t(u)=\exp _{\gamma(u)}(t \nu)$, are geodesics, filling a flat totally geodesic strip $(t \geq 0)$. Moreover, if $\gamma\left[u_0, u_1\right]$ is minimizing, then all $\gamma_t\left[u_0, u_1\right]$ are also minimizing.
Suppose this is true up to $t=\ell$ , and consider the function $$ f(r)=\max \left\{\left|x P\left(\exp _x(\ell+r) \nu\right)\right| \mid x \in S, \nu \in S N_x(S)\right\} $$ (1) He said that it's clear that $f$ is Lipschitz continuous, but I can't get understand it;
(2) The outline of the proof is "it is sufficient to check that if $(A)$ and $(B)$ hold for $0 \leq t \leq \ell$ for some $\ell \geq 0$, then they continue to hold for $0 \leq t \leq \ell+\varepsilon(\ell)$, for some $\varepsilon(\ell)>0$. ". However, can the process leads us to this conclusion is true for all $l\in (0,+\infty)$?
(3) By the way, the existence of such a distance nonincreasing retraction of $M$ onto $S$ is due to Sharafutdinov. Can anyone provide any references in English? Thanks in advance.
