The time evolution of these PDEs seeks to minimize a cost functional, which is an increasing function of the directional curvature magnitude of the image intensity function.

To prevent this maloperation,negative power directional unit can be used for judging the fault between internal and external,which is a way to prevent turn-to-turn faults protection maloperat.

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.

Based on the idea of integrable system,taking advantage of matrix model for 3-dimensional Minkowski space L 3,the integrability and curved suface deformation of negative curvity in L 3 were studied.