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

Let $\mathcal {M}$ be a Riemannian manifold, $p \in S \subset \mathcal{M}$ and $\delta>0$. 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 $r$.

$ \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 $r$.

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

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

Question 1: 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 $u \in \B(p,r)$. 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 $r$.

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

Let $\mathcal {M}$ be a Riemannian manifold, $p \in S \subset \mathcal{M}$ and $\delta>0$. 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 $r$.