I am actually in the resolution of the problem Show that $\Delta$ is diffeomorphic to X, so $\Delta$ is a manifold if $X$ is. - "Differential topology" of Guillemin and Pollack (my own question), and I was wondering if a subset of a smooth manifold is itself smooth manifold (submanifold).
Assume $X$ is a manifold and we have a smooth structure on the product manifodld $X×X$ , does that make $Δ$ a smooth submanifold? I already know that the answer is simply no, but is there sufficient conditions that we would provide an affirmation to this question?
I found the answer at my question. ''Open subsets of a smooth manifold are smooth manifolds. ''
reference : http://www-personal.umich.edu/~wangzuoq/635W12/Notes/Lec%2002.pdf