Let $G$ be a compact Lie group and $H$ a closed subgroup of $G$. Consider the homogeneous space $G/H$ with some $G$-invariant metric, and let $\sec$ denote the sectional curvature function associated to this metric.
Question: If $\mathrm{rank}(G)-\mathrm{rank}(H)\geq l$, is it true that there is an $l$-dimensional subspace tangent to $G/H$ on which $\sec\leq 0$?
Or perhaps, under what conditions is there an affirmative answer?
Berger proved that if $G/H$ has $\sec>0$, then $\mathrm{rank}(G)-\mathrm{rank}(H)\leq 1$. So the answer to the above question is "yes" if $l=2$.
If $G/H$ is assumed to be normal homogeneous, i.e. the metric is induced by a bi-invariant metric on $G$, then sectional curvature is given by $$\sec(X,Y)=\frac{1}{4}\big|[\hat{X},\hat{Y}]^\mathfrak{m}\big|^2+\big|[\hat{X},\hat{Y}]^\mathfrak{h}\big|^2,$$ where $\mathfrak{m}$ is the orthogonal complement of $\mathfrak{h}$ in $\mathfrak{g}$, and $\hat{X},\hat{Y}$ are the lifts of $X,Y$ to $\mathfrak{m}$. Is the answer to the above question also "yes" in this case?
Edit - Partial answer to last question: In a paper by Spatzier and Strake, they show that $\mathrm{rank}(G/H)\geq \mathrm{rank}(G)-\mathrm{dim}(H)$ for normal homogeneous spaces.