The Stiefel manifold is defined as follows. \begin{equation} V_{k}(\mathbb{C}^{n}) = \big\{ X \in \mathbb{C}^{n\times k} | X^{*} X = I_{k} \big\} \end{equation} where $n \geqslant k \geqslant 1$.
According to the pulic knowledge on Wikipedia on "Steifel manfiold", the dimension of manifold is given by \begin{equation} \operatorname{dim} V_{k}(\mathbb{C}^{n}) = 2nk -k^{2} \;. \end{equation}
Actually, the above dimension is only correct when $V_{k}(\mathbb{C}^{n})$ is considered a real manifold embedded in a $2nk$-dimensional real vector space $\mathbb{C}^{n\times k}$.
More naturally, the vector space $\mathbb{C}^{n\times k}$ should be considered as complex rather than real, and the dimension of the complex vector space $\mathbb{C}^{n\times k}$ is $nk$ rather than $2nk$.
If the manifold $V_{k}(\mathbb{C}^{n}) $ is considered as complex rather than real, the dimension, given by Wikipedia, I contend, is not correct.
I can not find useful information on the complexification of $V_{k}(\mathbb{C}^{n})$.
I also content, for example, when $k=1$, there is not such a thing as the complex Stiefel manifold, only real Stiefel manifold exists.
Under what condition, a complex Stiefel manifold exists?
What is the original source giving the resulf $\operatorname{dim} V_{k}(\mathbb{C}^{n}) = 2nk -k^{2}\,$? Is there anybody who can help?