Suppose $N$ is a closed submanifold in a Riemannian manifold $(M, g)$. For $q \in M \setminus N$, consider the sphere $$S = S\big(q, d(q,N)\big) = \{x \in M | d(q, x) = d(q, N)\},$$ where $d(q,N) = \min \{d(q, y) | y \in N\}$ is the distance from $q$ to $N$. Clearly, $S \cap N$ consists of precisely those points of $N$ where the distance $d(q,N)$ is attained.
Now, it seems very intuitive to me that if $d(q, N)$ is sufficiently small, the intersection is a singleton. My reasoning is that for $d(q,N)$ sufficiently small, the sphere is "too much curved", compared to the (compact) submanifold $N$, which makes sure that the sphere can only "touch" $N$ at a single point.
Is my reasoning remotely correct? If so, could someone please point out how to make this rigorous, or maybe point to any reference?
Thanks in advance. Cheers!
EDIT: I overdid it. As @ChesterX points out, it's much simpler than that. For each $p \in N$, there is an $\epsilon > 0$ such that $\exp_N$ is a diffeomorphism on a neighborhood of $(p,0)\in \Sigma^\epsilon$. By the compactness of $N$, there is a uniform $\epsilon$ such that $\exp_N$ is a diffeomorphism on $\Sigma^\epsilon$.
The longwinded discussion below is how to get a lower bound on $\epsilon$ in terms of the second fundamental form of $N$ and the Riemann curvature of $M$.
This is true, but the proof is somewhat involved. The first observation is that if a geodesic from $N$ to $q$ is minimal, then the geodesic is orthogonal to $N$. For each $p \in N$, let $\Sigma^\epsilon_p \subset T_pM$ be the set of tangent vectors orthogonal to $T_pN$ with length less than $\epsilon$. Let $$\Sigma^\epsilon = \cup_{p\in N} \Sigma^\epsilon_p.$$ Define the map \begin{align*} \exp_N: \Sigma^\epsilon &\rightarrow M\\ v &\mapsto \exp_p v\text{ for each }v \in \Sigma^\epsilon_p. \end{align*} It now suffices to show that for $\epsilon$ sufficiently small, this map is a diffeomorphism. The proof necessarily involves the second fundamental form of $N$ and the curvature of $M$, for exactly the reason you give.
The rest of the proof involves first gettng a lower bound for $\epsilon$ such that that $\exp_N$ is a diffeomorphism in a neighborhood in $\Sigma^\epsilon$ of $p \in N$. I suggest trying to work out the proof when $M$ is Eulicdean space.
In general, this involves estimating the differential of $\exp_N$, most easily done using Jacobi fields. The second fundamental form of $N$ and the sectional curvature of $M$ appear when analyzing the Jacobi equation. Compactness of $N$ then implies a uniform lower bound of $\epsilon$ for every $p \in N$.
I don't know if there is a differential geometry book that works this all out, but a paper that addresses this is a classic one by Heintze and Karcher.