Rigidity for closed manifolds with positive curvature

Let M be an n-dimensional complete connected Riemannian manifold with sectional curvature sec(M) >= 1 and radius rad(M) > pi/2. In this article, we show that M is isometric to a round n-sphere if for any x is an element of M, the first conjugate locus of x is a single point and if M contains a geodesic loop of length 2 . rad(M). We also show that the same conclusion is true if the conjugate value function at any point of M is a constant function.