Retraction on non-compact Stiefel manifold

Question

I have been looking online for the retraction of the non-compact Stiefel manifold (where $k <m$): $R_*^{m \times k} = \{ M \in \mathbb{R}^{m \times k} : \operatorname{rk}(M)=k \}$ as seen here (table 4) which links to there (Example 4.1.5) or here (section 3.4). I have read it to be $R_X(\dot{X}) = X + \dot{X}$ where $X \in \mathbb{R}_*^{m \times k}$ and $\dot{X} \in \mathbb{R}^{m \times k}$ i.e. $\dot{X}$ is in the tangent space $T_X \mathbb{R}_*^{m \times k}$. But the retraction should give a full rank matrix ? How is adding any $m \times k$ matrix going to ensure it ?