$C^{\infty } $ diffeomorphism between 2 nonequivalent manifolds

Question

On R we have 2 atlases

$A_1$={${u\in \mathbb R ,p_1:x\to x }$}

$A_2$={${u\in \mathbb R ,p_2:x\to x^3 }$}

$M_i$ is $C^{\infty }$ manifold whose differentiable structure generated by $A_i$ where i=1,2

I know from book both atlases are not equivalent

as $p_1\circ p_2^{-1}=x^{1/3}$ is not differemorphism

I wanted to know which is $C^{\infty }$ diffeomorphism $f:R \to R$ from M1 to M2.

Any Help will be appreciated

Answer 1

If you have any two smooth bijections $q_i : \mathbb R \to \mathbb R$, then you get two atlases $\mathcal A_i = \{ q_i\}$ determining two differentiable manifolds $M_i$.

But $f = q_2^{-1} \circ q_1 : M_1 \to M_2$ is smooth because $q_2 \circ f \circ q_1^{-1} = id$, similarly $g = q_1^{-1} \circ q_2 : M_2 \to M_1$ is smooth, and $f, g$ are inverse to each other. Thus $f$ is a diffeomorphism. In your example you have $q_1 = id$ and $q_2(x) = x^3$, thus $f(x) = x^{1/3}$.

Remark: A smooth bijection on $\mathbb R$ is a continuous bijection. It is easy to show that continuous bijections on $\mathbb R$ are homeomorphisms.