According to the idea and theorem,this paper describes the algorithm of creating charts for two-dimensional Riemannian manifolds and presents the definitions of transition and blend functions.

It is well-known that there is a unique vertex on rotating parabolic surface in three-dimensional Euclidiean space,the paper generalizes the concept of vertex to a complete noncompact Riemannian manifold with nonnegative curvature.

The paper discusses the structure of the soul in a complete noncompact Riemannian manifold M with nonnegative curvature,and proves that if the soul of the manifold is unique,then the soul actually degenerates to a pole.

The parallel properties of the rays in a complete noncompact Riemannian manifoldM were discussed in this paper, It is proved that the Busemann functions corresponding to any given two parallel rays are just the same as each other in the sense of H.

Delaunay triangulation and Voronoi diagrams for Riemannian manifolds

The studies of differential manifolds and their applications are motivated to the active fields with applications of Riemanian manifolds and Sub-Riemannian manifolds in Control Theory,Dynamics Theory,Gauge Fields,etc.

Estimations of the moments of the hitting time by Brownian motions on general Riemannian manifolds are also obtained.