Ricci Curvature And Boundedness Of Complete Riemannian Manifold

Myers'theorem is a classical theorem in Riemannian geometry. The theorem was proven by Myers in 1941, it states that if a n-dimensional complete Riemannian manifold with positive Ricci curvature lower bound (n-1)k2, then its diameter is at mostπ/k. The weaker result, due to Bonnet, has the same conclusion but under the stronger assumption that there is lower bound of sectional curvature.Moreover, if a n-dimensional complete Riemannian manifold of sectional curva-tures be bounded below by k(k>0) and its diameter is equal toπ/k, then the manifold is isometric to a n-dimensional sphere with radius equal to k. The result is called Topono-gov's maximum diameter theorem. For the Ricci curvature case there is also Cheng's maximum diameter theorem.Myers'theorem also holds for the universal cover of the n-dimensional complete Riemannian manifold of positive Ricci curvature lower bound, in particular both the manifold and its universal cover are compact, so the cover is finite-sheeted and the manifold has finite fundamental group.In this thesis, we give a survey of Myers' type theorem. We will relax the con-ditions of Myers theorem, exploring the condition satisfied by lower bound of Ricci curvature which can still guarantee the boundedness of n-dimensional complete Rie-mannian manifolds. We will consider the case with nonnegative Ricci curvature, the case with negative lower bound of Ricci curvature, and the case with Ricci curvature unbounded. We discuss in these cases the constraints to ensure the boundedness of a complete Riemannian manifold.