$
\newcommand{dist}{\operatorname{dist}}
\newcommand{B}{\mathbb{B}}
$


Let $\mathcal {M}$ be a Riemannian manifold, $p \in S \subset \mathcal{M}$ and $r>0$. Denote $S\_{r} := S \cap\B(p,r)$.


**Question 1:** For small values of $r$, Is there a relation similar to the following
$$
\dist(\exp\_p^{-1}(u);\exp\_p^{-1}(S\_r)) \leq C(r)\dist(u;S\_r) \tag{1}
$$ 
for every $u \in \B(p,r)$.
Note that $C(r)$ may be related to the curvature.


I know that the inequality (1) holds with $C(r) = 1$, for a Hadamard manifold (since the exponential function has non-expansion property for a Hadamard manifold).


**Question 2:** Is there an asymptotic relation between $\dist(u;S\_r)$ and LHS of (1) for small values of $r$.



Relation between a distance function and normal coordinations
Lemma 3.24 at page 87 of Alexander Grigor'yan's "Heat Kernel and Analysis on Manifolds" says that (I reformulate it slightly)



> 
> For any point $p \in M$ and chart $(U', h)$ around $p$ there exist a $U \subseteq U'$ and a constant $C \ge 1$ such that for all $x,y \in U$ we have
> 
> 
> $$\frac 1 C \| h(x) - h(y) \| \le d(x,y) \le C \| h(x) - h(y) \| \ .$$
> 
> 
> 


Taking $h = \exp\_p ^{-1}$ and $v \in S\_r$ you get


$$\frac 1 C \| \exp\_p ^{-1} (u) - \exp\_p ^{-1} (v) \| \le d(u,v) \le C \| \exp\_p ^{-1} (u) - \exp\_p ^{-1} (v) \| \ ,$$


whence by taking $\sup \_{v \in S\_r}$ one gets


$$\frac 1 C \| \exp\_p ^{-1} (u) - \exp\_p ^{-1} (S\_r) \| \le d(u,S\_r) \le C \| \exp\_p ^{-1} (u) - \exp\_p ^{-1} (S\_r) \| \ ,$$


the first inequality being what you are asking for.


By looking at Grigor'yan's proof, it seems to me that for small enough $U$ and $r$ one may choose $C$ independent of $p$ and $r$.


