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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?

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Is this inequality sharp? –  Deane Yang Aug 9 '11 at 13:58
    
Deane: It seems to be. According to the paper below, $CP^N$ saturates the inequality: $$ $$ ams.org/journals/proc/1970-026-04/S0002-9939-1970-0270304-5/… –  José Figueroa-O'Farrill Aug 9 '11 at 14:11
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This inequality is used in very interesting constructions of symplectic manifolds via "fat bundles". See Theorem 4.7 in Kedra-Tralle-Woike, arXiv:1004.3699. Similar ideas are used in Fine-Panov, arXiv:0802.3648 (and other papers by Fine-Panov) and go back to Alexander Reznikov's paper "Symplectic twistor spaces". –  Dan Fox Sep 11 '11 at 10:51

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