When $F = \mathbb{R}, \mathbb{C}$ or $\mathbb{H}$, there are fibrations $$O(k,F)\rightarrow V_k(F^n)\rightarrow G_k(F^n)$$ where $V_k(F^n)$ are Steifel manifolds and $G_k(F^n)$ are Grassmannians. When $k=1$ these reduce to the Hopf fibrations $$S^{d-1}\rightarrow S^{dn-1}\rightarrow FP^{n-1}$$ where $d=\text{dim}F$.

If $S^{dn-1}$ are given round metrics, people make a big deal about the metrics on $FP^{n-1}$ that make the above Riemannian submersions.

You can also describe the metrics on $FP^{n-1}$ with distances between subsets of the round sphere $S^{dn-1}$ that correspond to lines through the origin in $F^n$. This, from what I've read (and agree with), is also the natural metric to put on $G_k(F^n)$ where "lines" has been replaced by "$k$-planes".

I'm being tricked into believing that there should be some nice metric on $V_k(F^n)$ that makes the map $V_k(F^n)\rightarrow G_k(F^n)$ Riemannian and that I must not have been listening when people were making an equally big deal about these metrics on $G_k(F^n)$ as they do for $FP^n$.

**Question**: What goes wrong? If nothing, why is this not as interesting as the geometry of the Hopf fibrations? If equally interesting, where is a good source to learn more?