Cube with smooth structure

Question

My question is:

Can the cube $C=\{x\in \mathbb{R}^{n}; \max\{|x_{i}|; 1\leq i \leq n\} = 1\}$ be endowed with a smooth differentiable structure? If not, how can I justify that?

I've been trying to prove that it is a submanifold of $\mathbb{R}^{n}$, but I didn't succeed. I also tried to find a way to see that it is not true (once I'm in doubt), but I couldn't either.

Thanks

Answer 1

Nice question. The answer is that it can be endowed with a smooth differentiable structure, but it does not naturally inherit one from $\mathbb{R}^n$. To give it a differentiable structure, you could for example pick a homeomorphism from the cube to a sphere, and use it to endow the cube with a differentiable structure.