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) > π/2. In this article, we show that M is isometric to a round n-sphere if for any x ∈ 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. This work was done while the author was visiting MPI for Mathematics in Leipzig, Germany. The author is very grateful to MPI for Mathematics in Leipzig for its hospitality and CAPES.