Full-row-rank matrix manifold ($X_{m,n}$)

The real $m$ by $n$ matrices of matrix rank $m\le n$ form an open smooth manifold $X_{m,n}\subseteq {\mathbb{R}}^{mn}$. Indeed, it is the union of the open sets on which a chosen $m$ by $m$ minor has nonzero determinant, so its dimension is $mn$.