In Boothby's "An introduction to Differentiable Manifolds ...", page 69, Remark 4.2:
I am unable to discover two cubes of same side length, as asked for. I shall feel very thankful if anyone give some hint.
In Boothby's "An introduction to Differentiable Manifolds ...", page 69, Remark 4.2:
I am unable to discover two cubes of same side length, as asked for. I shall feel very thankful if anyone give some hint.
You are not actually required to describe such cubes. What the remark tells you is that, up to composition by some diffeomorphism, you may assume that $\phi(U)$ and $\psi(V)$ are cubes of the same side length. This gives you a way to locally visualise the image of $N$ in $M$.