SPHERE THEOREM FOR MANIFOLDS WITH POSITIVE CURVATURE
Glasgow Mathematical Journal
In this paper, we prove that, for any integer n ≥ 2, and any δ > 0 there exists an (n, δ) ≥ 0 such that if M is an n-dimensional complete manifold with sectional curvature K M ≥ 1 and if M has conjugate radius ρ ≥ π 2 + δ and contains a geodesic loop of length 2(π − (n, δ)) then M is diffeomorphic to the Euclidian unit sphere ޓ n . 2002 Mathematics Subject Classification. 53C20, 53C21. Introduction. One of the fundamental problems in Riemannian geometry is to determine the relation between
... topology and the geometry of a Riemannian manifold. In this way the Toponogov's theorem and the critical point theory play an important role. Let M be a complete Riemannian manifold and fix a point p in M and define d p (x) = d ( p, x). A point q = p is called a critical point of d p or simply of the point p if, for any nonzero vector v ∈ T q M, there exists a minimal geodesic γ joining q to p such that the angle (v, γ (0)) ≤ π 2 . Suppose M is an n-dimensional complete Riemannian manifold with sectional curvature K M ≥ 1. By Myers' theorem the diameter of M is bounded from above by π. In  Cheng showed that the maximal value π is attained if and only if M is isometric to the standard sphere. It was proved by Grove and Shiohama  that if K M ≥ 1 and the diameter of M diam(M) > π 2 then M is homeomorphic to a sphere. Hence the problem of removing homeomorphism to diffeomorphism or finding conditions to guarantee the diffeomorphism is of particular interest. In  C. Xia showed that if K M ≥ 1 and the conjugate radius ρ(M) of M is greater than π/2 and if M contains a geodesic loop of length 2π , then M is isometric to ޓ n .