Define $f:S^n\to S^n$ by $x\to -x$. Prove this is a diffeomorphism.
My attempt: My definition of a smooth map between two manifolds $M,\,N$ is that a map $g:M\to N$ is smooth iff the maps: $$\psi_i\circ g\circ \phi_j^{-1}$$ is smooth (i.e it is smooth in local coordinates) for some maps $\psi_i:V\to\mathbb{R}^n$ and $\phi_j:U\to\mathbb{R}^n$ in the smooth atlases of $N$ and $M$ respectively.
The fact that this map is a hoemomorphism is clear. However, I'm stuck with proving it is smooth. I fixed some maximal smooth atlas $\{(\phi_i,V_i)\}$. For every $x\in\phi_i(V_i\cap V_j)$: $$\phi_j\circ f\circ \phi_i^{-1}(x)=\phi_j(-\phi_i^{-1}(x))$$ I feel like this is obviously smooth but can't seem to find a formal way to prove it. I thought maybe it's worth showing that $\psi_i(x)=\phi_i(-x)=(-\phi_i(x)^{-1})$ is in the maximal atlas, but wasn't able to, or find specific maps that I can work with – but I don't really know any specific maps for $S^n$.
Any help would be appreciated.
Best Answer
$S^n$ is a submanifold of $\mathbb R^{n+1}$. The map $x \mapsto -x$ is clearly a diffeomorphism on $\mathbb R^{n+1}$. It maps $S^n$ to itself. Therefore it is a smooth map on $S^n$ with a smooth inverse.
All you need to know here are basic properties of submanifolds.