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 This, from what I've read (and agree with), is also the natural metric to put on $G\_k(F^n)$ where "lines" has been replaced by "$k$-planes".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?



Geometry of Hopf fibrations and the fibration of Steifel Manfiolds over Grassmannians
You are right, this is true. You start with the invariant metric on $O(k,F)$. Then the relevant subgroups act isometrically, thus you get Riemannian submersions. 
The Grassmann manifolds are symmetric spaces, but the Stiefel manifolds are only homogeneous.
A very good source which also gives explicitly all geodesics and even the geodesic distance
on Grassmannians is:


* Y. A. Neretin, On jordan angles and the triangle inequality in
grassmann manifold, Geometriae Dedicata, 86 (2001).


