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

Abstract

Let E be a smooth bundle with fiber an n-dimensional real projective space RP n . We show that, if every fiber carries a pointwise strongly 1/4-pinched Riemannian metric that varies continuously with respect to its base point, then the structure group of the bundle reduces to the isometry group of the standard round metric on RP n .