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

Abstract

Let E E be a smooth bundle with fiber an n n -dimensional real projective space R P n \mathbb {R}P^n . We show that, if every fiber carries a positively curved pointwise strongly 1 / 4 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 R P n \mathbb {R}P^n .