I know that we can refered to the question How to show that the diagonal of $X\times X$ is diffeomorphic to $X$?. I have the same question with an answer, and I needed that someone tell me if it is good or not (why?)
Here's the question :
The diagonal $\Delta$ in $X \times X$ is the set of points of the fom $(x,x)$. Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is.
Solution : First, let $ϕ:X→Δ$ $ϕ:x↦(x,x)$. This is a smooth function, as it extends easily to $Φ:R^N→R^N×R^N$, $Φ(x)=(x,x)$ which is linear. Its inverse, $ϕ^{−1}(x,x)=x$ is a restriction of the (smooth) projection function.