Topological Structure Of Riemann Manifold With Positive Curvature

The main work of this paper is to study sphere theorem and differentional sphere theorm in Riemannian manifolds with positive curvature.we can use the tools such as volume comparison theorem、Toponogov triangle comparison theorem、cosine theorem of hyp erbolic geometry.The main results as following:1.let M be an2n-dimensional simple connected compact Riemannian manifolds,there exists an positive number η depending only on dimension,if M has sectional curvature0<Km≤1, Ricci curvature and volume,then M is homeomorphic to unit sphere S2n.Theorem can be proved by contradiction.assume that the condition of the theorem is established,but M is not homeomorphic to unit sphere S2n. First, we can prove that the diameter of this manifold di satisfy Second, we can prove by comparison theorem and cosine theorem, which contract to the first step,then M is homeomorphic to unit sphere S2n. This theorem implies if we restrict the sectional curvature and loose the volume condition, the sphere theorem still established.2. Let M be an n dimension complete Riemannian manifold, for any integer n≥2, There exists ε(n)>0depending only on n such that for any,if M has radial curvature Ricci curvature conjugate radius and if M contains a geodesic loop of length2(π-ε),then M is differomorphic to the unit sphere S".we can prove it by theorem: differomorphic to the unit sphere Sn.. it suffices to show that we can prove by following lemma.This theorem tells us that M is still differomorphic to the unit sphere Sn when the curvature condition weakened. Following known theorem,we can get following corollary:(1)Giveni0>0,there exists an ε>0depending only on n,i0such that if M is an n-dimension complete Riemannian manifolds with,then M is homeomorphic to Sn.(2) Given k∈R+, v>0,there exists an ε>0depending on n,k,v such that if M is an n-dimension complete manifold with and then M is homeomorphic to Sn.