Singularities of the Curve Shortening Flow in a Riemannian Manifold

In this paper,the curve shortening flow in a general Riemannian manifold is studied,Altschuler’s results about the flow for space curves are generalized.For any n-dimensional(n ≥ 2)Riemannian manifold(M,g) with some natural assumptions,we prove the planar phenomenon when the curve shortening flow blows up.