# The relationship between the tubular neighbourhoods of two diffeomorphic manifolds

I'm a beginner of this complex area and want to use the differential geometry as a tool to solve some control problems. So my statement might be a little bit inaccurate...I will try my best.

There are two compact, smoothly embedded submanifolds $$M$$ and $$N$$ $$\subset\mathbb{R}^m$$. M and N are diffeomorphic with a diffeomorphism $$f$$ from $$M$$ to $$N$$.

If there is a point $$x$$ on the tubular neighborhood of $$M$$, which is defined $$U_M^\delta$$. The retraction projection of x is defined $$rm(x)$$ so that $$rm(x)\in M$$ and $$x-rm(x)$$ always on the normal space of $$M$$ at point $$rm(x)$$.

In the same way, there is a point $$y$$ on the tubular neighborhood of $$N$$, which is defined $$U_N^\delta$$. The retraction projection of $$y$$ is defined $$rn(y)$$ so that $$rn(y)\in N$$ and $$y-rn(y)$$ always on the normal space of $$N$$ at point $$rn(y)$$.

(I am not clear whether it's right that I use the same symbol $$\delta$$ for both tubular neighborhood. I just want to define some symbols to express the tubular neighborhood)

My Question:

Could I augment the domain and codomain of $$f$$ from $$f:M\to N$$ to $$f:U_M^\delta\to U_N^\delta$$ so that for a point $$x$$ on the $$U_M^\delta$$, $$f(x)$$ will be on the $$U_N^\delta$$? And for the retraction projection of $$x$$ on the $$M$$, which is $$rm(x)$$, it must have $$f(rm(x))\in N$$. Is this true that $$f(rm(x))=rn(f(x))$$? If so, could you provide me a proof or recommend me some textbooks about the proof of this equality? Thanks.

You cannot extend a diffeomorphism $$f: M \rightarrow N$$ to any tubular neighbourhoods of $$M$$ and $$N$$ but as $$M$$ and $$N$$ are compact, you can extend $$f$$ to a diffeomorphism of some suitable smaller tubular neighbourhoods of $$M$$ and $$N$$. This diffeomorphism would satisfy your properties. The proof is based on extending the diffeomorphism to the tubular neighbourhood of $$M$$ and then showing that the image of this extended $$f$$ forms a tubular neighbourhood of $$N$$ provided the tubular neighbourhood of $$M$$ is sufficiently small.
Edit: The proof I had in mind uses the fact that $$f$$ induces a diffeomorphism of the tangent bundles $$f^*:TM \rightarrow TN$$, one then extends this to the normal bundles and then uses the exponential map to get a diffeomorphism of the tubular neighbourhoods. Kobayashi & Nomizu sounds like the obvious source but I haven't checked. Thinking about it I wonder whether one needs some condition of manifolds being parallelizable for this to work, $$\mathbb{R}^n$$ is parallelizable but I'm not fully sure whether that is sufficient.
• Hi, I'm still a little confused about the proof. Could you explain it in more detail or where can I find such proof? In addition, the diffeomorphism $f$ is some affine function, like $f:M\to N, rm\mapsto\frac{rm-a}{b}$, where $a\in\mathbb{R}^m$ and $b\in\mathbb{R}$. It likes that I can change the position and size(e.g. from a large circle to a small circle) or give additional definition of $b$ so that can change the shape(e.g. from a circle to an ellipse). Thanks Commented Mar 10, 2020 at 23:04