The Geodesics In Limit Spaces With Ricci Curvature Bounded Below

This paper is divided into three parts.we first introduce the Toponogov-type angle comparison theorem of a small geodesic triangle,which is proved by Wei Guofang,under the condition that the complete Riemannian manifold with the lower bound of the conjugate radius and Ricci curvature.Secondly,we introduce some relevant concept and proof of the toponogov-type hinge comparison theorem of a small geodetic triangle given by Dai Xianzhe and Wei Guofang under the condition that the complete Riemannian manifold with the lower bound of the conjugate radius and Ricci curvature.Finally,using the method that to prove the minimal geodesics in the limit space of a subclass manifolds with lower bounds on section curvature can not branch by K.Grove and P.Petersen.we discussed the minimal geodesics in the limit space of a subclass manifolds with lower bounds on Ricci curvature(i.e.and conjugate radius)also can not branch.we are concerned with limits of sequences of manifolds with lower bound on Ricci curvature and conjugate radius and Toponogov-type comparison theorem to prove that the minimal geodesic in limit space is the limit minimal geodesics.