In the paper, we used some types of asymptotically nonnegative curvature to get some conditions of complete noncompact Riemannian manifold which is nonparabolic.

By Buseman function,the author discusses the geometry and topology of a complete manifolds with quadratically nonnegatively curved infinity.

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 gathers some results in Riemannian manifolds,including in complete geodesics without conjugate points,the geometric struture of a manifold with nonnegative curvature,the topology of a manifold with nonnegative Ricci curvature and some properties of Busemann function etc.

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 topology of complete manifolds with nonnegative Ricci curvature and large volume growth;

For an open complete Riemannian manifold with nonnegative Ricci curvature,the present paper discusses the relation between the topology and the volume growth.

By comparing the volume growth order of the manifold itself to that of its universal covering space, the paper proves that every three-dimensional with nonnegative Ricci curvature and (1+δ)-order volume growth in strict sense must be contractible provided that its universal covering is finite.