In the Appendix 5 to Ballmann, Gromov and Schroeder's Manifolds of nonpositive curvature, Schroeder mentions that for a rank one symmetric space of noncompact type, $-1 \leq K \leq -\frac{1}{4}$ (possibly after rescaling the metric, of course) where $K$ is sectional curvature.
I was not able to find a reference for this fact, so if anyone knows one, that would be great. I would also be fine with a reference for the fact that the sectional curvature for a rank one symmetric space of compact type is $\frac{1}{4} \leq K \leq 1$.
I suppose since there's only a few families of noncompact type, rank one symmetric spaces, one could compute it case by case, but maybe there's a general proof someone could point me to. Thanks!
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$\begingroup$Good question. These results are surprisingly hard to find. (Books by Helgason and by Eberlein, which are standard references for geometry of symmetric spaces, do not contain the results.)
- For rank 1 compact symmetric spaces (CROSSes), quater-pinching is proven in
Chavel, Isaac, On Riemannian symmetric spaces of rank one, Adv. Math. 4, 236-263 (1970). ZBL0199.56403.
- For negatively curved symmetric spaces, quarter-pinching is proven in
Heintze, Ernst, On homogeneous manifolds of negative curvature, Math. Ann. 211, 23-34 (1974). ZBL0273.53042.
Most likely, one can find even earlier references (probably by sifting through papers by Elie Cartan.)
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