The relationship between the tubular neighbourhoods of two diffeomorphic manifolds

Question

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.

Answer 1

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.