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Let $E$ be a smooth bundle with fiber an $n$-dimensional real projectivespace $\mathbb{R}P^n$. We show that, if every fiber carries a positively curvedpointwise strongly $1/4$-pinched Riemannian metric that varies continuouslywith respect to its base point, then the structure group of the bundle reducesto the isometry group of the standard round metric on $\mathbb{R}P^n$.

Bundles with even-dimensional spherical space form as fibers and fiberwise quarter pinched Riemannian metrics