I'm trying to show that set subset of matrice of X in $M_{n}(R)$ such that X is idempotent, symmetric and with Tr(X) =1 is a submanifold of $M_{n}(R)$.
My idea is to consider the set $ \{(X,(0,0,0)) \}$ as the graph of the function $f(X) \longrightarrow (X^2-X, X^t-X, tr(X)-1)$
The fact is that submanifold is kinda a new notion to me and I'm not 100% comfortable with it. So I'm not sure that what I've done is correct, or if there is a more efficient way to do it.
I've tried to construct submersions and immersions for the caracterisation of submanifold, but I didn't success to show that the map I constructed were injection or surjective.
Thank you very much for some explenation or alternative solution.