What is known about the "stickiness" of a smooth manifold to its tangent space?

Question

Consider an $m$-dimensional smooth manifold $M$ embedded in $n$-dimensional real Euclidean space $\mathbb{R}^n$ ($n>m$). Consider a point $x \in M$ of the manifold, and for simplicity, choose the coordinate axes of the Euclidean space $\mathbb{R}^n$ such that $x$ is in the origin ($x = 0$).

Consider the tangent space $T_0 M$ of the manifold at this point, and think of this tangent space as a linear subspace of $\mathbb{R}^n$. Consider a specific line in the tangent space, $\gamma(t) = v t$ with $v \in T_0 M$ being a unit vector in the tangent space. As we imagine a particle traveling along this line, we can also imagine its shadow $s(t) \in M$ on the manifold $M$ traveling along; the shadow $s(t) \in M$ of the particle is defined as the closest point of the manifold $M$ to the particle. Hence we can define the distancing function of the particle from the manifold as $d_v(t) = |vt-s(t)|$.

Question 1: Is the distancing function $d_v(t)$ an analytic function of t? I will assume it is, and probably it is easy to prove this.

Call the lowest non-vanishing order $r_v \in \{1,2,\dots,\infty\}$ of $d_v(t)$ the order of distancing along $v$. The higher $r_v$, the more the manifold $M$ "sticks" to the tangent space $T_0M$ along that direction. Hence, it makes sense to use the word stickiness as an alternative to the order of distancing. In fact, since we consider directions in the tangent plane, I suspect that $r_v = 1$ is not possible, hence $r_v \in \{2,3,\dots,\infty\}$.

Up to now, for each unit vector $v \in T_0M$, we have assigned an integer $r_v$. In other words, we defined a map $r$ from the unit sphere of the tangent space to the set $\{2,\dots,\infty\}$. Call this map $r$ the stickiness structure of the manifold at the point under consideration. Two simple examples of this map are as follows.

Example 1: Take an arbitrary point on the two-dimensional sphere embedded in $\mathbb{R}^3$. I suspect that the stickiness in any direction of the tangent plane is 2.

Example 2: Take an arbitrary point on the two-dimensional cylinder embedded in $\mathbb{R}^3$. I suspect that the stickiness along the axial direction is $\infty$, whereas the stickiness in any other direction is 2.

Question 2: Is this "stickiness structure" defined and studied in the literature of smooth manifolds, either in general, or for specific examples?

Question 3: Is there a "nested linear structure" within the stickiness structure, in general? In more detail: Take two directions (unit vectors) $v$ and $w$ in the tangent space $T_0M$, which are path-connected according to the stickiness structure: they have the same stickiness, $r_v = r_w$, and there is a continuous path connecting them such that all unit vectors along that path have the same stickiness. Is it true that all directions in the subspace spanned by $v$ and $w$ have stickiness $r_v$ or higher? Is this true if we demand only $r_v = r_w$, but do not demand the path-connectedness?

Remark: Question 3 is motivated by Example 2, the cylinder, where this nested linear structure is present (even without the path-connectedness).

Answer 1

This answer is more of an extended comment.

Unless I misunderstand something, this is only meaningful for embeddings, i.e. it is not an intrinsic invariant of a manifold. For example you can take an open plane segment as your manifold and then its "stickiness" is very large because it essentially coincides with its tangent space. But you can also embed the same (in the sense of diffeomorphic) plane segment as a kind of "spherical cap", and then its "stickiness" will be smaller because the tangent space only touches it. So this property depends rather explicitly on embeddings. I suppose you are aware of this, but I wanted to point it out because it's not mentioned anywhere in the question.

Otherwise, once again, unless I misunderstand the question, this concept seems to be covered by the concept of "jets of submanifolds".

If $M\subseteq\mathbb R^n$ is the given submanifold, then we can regard $T_0M$ as a submanifold of $\mathbb R^n$ as well. Then the vector $v\in T_0M$ selects a line in $T_0M$, let's call it $L$.

It seems to me that your concept of "stickiness" is the order of contactness of $L$ and $M$ at $0$, which is necessarily $\ge 1$.

In general, let $N$ be a manifold, let $M,M^\prime\subseteq N$ be submanifolds (not necessarily of the same dimension). Let $m=\dim M$ and $m^\prime=\dim M^\prime$. Let's say that $M$ and $M^\prime$ have contact order $0$ at $p\in N$ if $p\in M\cap M^\prime$. Suppose that $m^\prime\le m$. Then they have contact order $1$ at $p$ if they have the contact order $0$ at $p$ and in addition to that $T_pM^\prime\subseteq T_pM$.

Then we can say that $M$ and $M^\prime$ have the contact order $r$ at $p$ if they have contact order $0$ there and we have $T^r_pM^\prime\subseteq T^r_pM$ where $T^r_p$ is the $r$th order tangent space (i.e. $T_p^rM=J^r_{0,p}(\mathbb R,M)$).

I don't know any source that treats this when $m\neq m^\prime$, but for $m=m^\prime$ this defines the notion of "$r$-jet of an $m$ dimensional submanifold", which is a well-known concept.