# Definition and intuition of a tubular neighborhood of a submanifold

Let $$d\in\mathbb N$$, $$k\in\{1,\ldots,d\}$$ and $$M$$ be a $$k$$-dimensional embedded $$C^1$$-submanifold of $$\mathbb R^d$$ with boundary$$^1$$.

Now let $$T_xM:=\left\{v\in\mathbb R^d\mid\exists\varepsilon>0,\gamma\in C^1((-\varepsilon,\varepsilon),M):\gamma(0)=x,\gamma'(0)=v\right\}$$ denote the tangent space of $$M$$ at $$x$$ and $$N_xM:=(T_xM)^\perp$$ for $$x\in M$$.

I'm trying to understand the definition of a tubular neighborhood of $$M$$. I was only able to find definitions of this notion in a way more general setting which built up on several concepts I'm not familiar with. So, I'd like to find a simplified, but equivalent, definition for my present setting.

What I've understood is that one considers the space $$N(M):=\{(x,v):x\in M\text{ and }v\in N_xM\}$$ and the map $$E(x,v):=x+v\;\;\;\text{for }(x,v)\in N(M).$$

Now the usual definition of a tubular neighborhood $$U$$ of $$M$$ is that it is a (open?) neighborhood of $$M$$ in $$\mathbb R^d$$ (really a neighborhood of all of $$M$$?) such that $$U$$ is the diffeomorphic image under $$E$$ of an open subset $$V$$ of $$N(M)$$ with $$V=\{(x,v)\in N(M):\left\|v\right\|<\delta(x)\}\tag1$$ for some continuous function $$\delta:M\to(0,\infty)$$.

I really got trouble to understand this. The vector $$E(x,v)$$ is simply a vector originating from $$x$$ and pointing in the direction $$v$$. If $$M$$ is a circle in $$\mathbb R^3$$, I guess the intuition is that for each point $$x$$ on the circle all the $$v\in N_xM$$ built a ring around $$x$$ by "rotating" the normals around $$x$$. Doing so for all $$x$$ on the circle, we obtain a torus. Is this correct so far? All the pictures I've found have confused me.

And how would a tubular neighborhood of a sphere in $$\mathbb R^3$$ look like? For a sphere, the normal spaces are 1-dimensional ...

$$^1$$ i.e. each point of $$M$$ is locally $$C^1$$-diffeomorphic to $$\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$$.

If $$E_i$$ is a $$\mathbb R$$-Banach space and $$B_i\subseteq E_i$$, then $$f:B_1\to E_2$$ is called $$C^1$$-differentiable if $$f=\left.\tilde f\right|_{B_1}$$ for some $$E_1$$-open neighborhood $$\Omega_1$$ of $$B_1$$ and some $$\tilde f\in C^1(\Omega_1,E_2)$$ and $$g:B_1\to B_2$$ is called $$C^1$$-diffeomorphism if $$g$$ is a homeomorphism from $$B_1$$ onto $$B_2$$ and $$g$$ and $$g^{-1}$$ are $$C^1$$-differentiable.

Your intuition seems to be right. I like to describe it as an open disk bundle i.e. the fibre over the point $$p \in M$$ looks like $$\{p\} \times O^n$$ where $$O$$ is an open disk of dimension $$n$$ (which is also equal to the dimension of the normal bundle, or even fancier, the codimension of $$M$$). So you have a tube that encapsulates the submanifold.
If $$M$$ is a circle in $$\mathbb{R}^3$$ then you are right, its tubular neighbourhood is exactly an (open solid) torus i.e. $$S^1 \times O^2$$ so it's actually a trivial bundle.
Yes, all of $$M$$. Now let us consider the sphere $$S^2 \subset \mathbb{R}^3$$. You get essentially an (open) thickened sphere, you can think of it as $$S^2 \times O^1 \cong S^2 \times (0,1)$$. Note the dimension of the open disk you are working with, since the sphere is codimension $$1$$!
The general idea is that any submanifold can be identified with the zero section in its normal bundle, and in the normal bundle you have a little "wiggle room" in the normal direction around each point and gluing all these little wiggle rooms together give you an open neighbourhood of $$M$$ in the ambient manifold.
• Perhaps a slightly more interesting example for the OP to contemplate is a tubular neighborhood of $\Bbb RP^1\subset\Bbb RP^2$ (or, indeed, the normal bundle of $\Bbb RP^n\subset\Bbb RP^{n+1}$). Move on to $\Bbb CP^n$ eventually. :) Jul 21, 2020 at 17:03