ON THE GEOMETRY OF POSITIVELY CURVED MANIFOLDS WITH LARGE RADIUS

摘要

Let M be an n-dimensional complete connected Riemann-ian manifold with sectional curvature K_M ≥ 1 and radius rad(M) > π/2. For any x ∈ M, denote by rad(x) and ρ(x) the radius and conjugate radius of M at x, respectively. In this paper we show that if rad(x) ≤ ρ(x) for all x ∈ M, then M is isometric to a Euclidean n-sphere. We also show that the radius of any connected nontrivial (i.e., not reduced to a point) closed totally geodesic submanifold of M is greater than or equal to that of M.