Is the set $\{x \in \mathbb R^n : d(x, M) = c\}$ a smooth manifold for a small constant $c$ when $M$ is a smooth manifold embedded in $\mathbb R^n$?

Question

I am a beginner of differential geometry. Let $M$ be a smooth manifold embedded in $\mathbb R^n$ and consider the subset $$ S = \{x \in \mathbb R^n : d(x, M) = c \}, $$ where $d(x, M)$ denotes the distance between a point $x$ and the manifold $M$. I would like to know if $S$ is a smooth (at least $C^1$) manifold for sufficiently small $c > 0$. In one of the simplest cases, that $M$ is a (finite) line segment in $\mathbb R^3$, I think $S$ is the union of a cylinder of radius $c$ with its center line $M$ and two half spheres attached to the both ends of the cylinder, and $S$ is smooth.

Answer 1

Not necessarily; the most important issue is that two points of $M$ which are "far apart inside $M$" might actually be very close together in $\mathbb{R}^n$. For example, let $M$ be the image of the smooth embedding $f : \mathbb{R} \to \mathbb{R}^2$ given by $f(t) = (\cos(2\arctan(t)), \sin(2\arctan(t)))$; i.e. $M = S^1 \setminus \{(-1,0)\}$. For any $c > 0$, the set $\{x \in \mathbb{R}^2 : d(x,M) = c\}$ contains two points which have no open neighborhood homeomorphic to any Euclidean space.

This issue is obliterated if $M$ is compact; then the result does hold (with $S$ being smooth, assuming $M$ is smooth as stated in the question). There is a more general result called the "tubular neighborhood theorem" or the "$\varepsilon$-neighborhood theorem" which may be of interest if you care about cases where $M$ is not compact.