Let E be a smooth bundle with fiber an n-dimensional real projective
space RPn. We show that, if every fiber carries a positively curved
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 RPn.Comment: 11 pages, to appear in Proceedings of the American Mathematical
Societ