Geometrical Structure Of Differentiable Dynamical Systems On Riemannian Manifold

The theory of dynamical systems is the important parts of mathematics .In recent years, people have already believe that the geometrical structure has some important influence on dynamical systems. In the past, people used to take the local coordinate of manifold as the state variable of dynamical systems. But it cannot embody the influence of the geometrical structure. For example, the state function of dynamical systems is the same as on different manifolds.First of all, we take the tangent field as the state vector field of dynamical systems and defined a kind of differentiable dynamical systems on Riemannian manifolds, and we also gave the equation for the differentiable dynamical systems under a local coordinate system of Riemannian manifold. The second, we show that the geometrical structure of two-dimensional Riemannian manifold has some important influence on differentiable dynamical systems. The third, we discuss the ways to simplify the equation of differentiable dynamical systems. The last, we also discuss the particular points on some surfaces.