I'm interested in applications of the following estimate of Berger on the Riemann curvature tensor: Let $(M,g)$ be a Riemannian manifold of dimension $n \geq 4$, let $p \in M$, and assume that the sectional curvature $K(\pi)$ at $p$ lies in $ [\lambda,\Lambda]$ for all $2$-dimensional subspaces $\pi \subset T_pM$. Then for any orthonormal collection $\{e_1,e_2,e_3,e_4\}$ in $T_pM$, the Riemann curvature tensor $R(.,.,.,.)$ satisfies
$|R(e_1,e_2,e_3,e_4)| \leq \frac{2}{3}(\Lambda-\lambda).$
This is obtained by using the symmetries of the curvature tensor and the first Bianchi identity. One nice application is that pointwise strict quarter-pinching of sectional curvature implies positive isotropic curvature.
Does anyone know of other striking applications?
