Rigidity of compact rank one symmetric spaces

We consider rigidity properties of compact symmetric spaces X with metric $$g_0$$ g 0 of rank one. Suppose g is another Riemannian metric on X with sectional curvature $$\kappa $$ κ bounded by $$0 \le \kappa \le 1.$$ 0 ≤ κ ≤ 1 . If g equals $$g_0$$ g 0 outside a convex proper subset of X , then g is isometric with $$g_0.$$ g 0 . We also exhibit examples of surfaces showing that the nonnegativity of the curvature is needed. Our main result complements earlier results on other symmetric spaces by Gromov and Schroeder–Ziller.