EVOLUTION METRICS AND GEOMETRIC VECTOR FIELDS

摘要

This paper opens the study of the metric flows and Riemannian metric dynamics generated by the covariant derivative of geometric vector fields on Riemannian manifolds. Section 1 recalls some basic geometric vector fields on Riemannian manifolds. Section 2 studies the evolution pdes satisfied by the geometric entities induced by the flow of Riemannian metric. Section 3 finds the evolution pdes induced by Riemannian metric dynamics. Section 4 and Section 5 give the equilibrium points and the steady states of some geometric evolutions. Section 6 and Section 7 study the metric flow or the metric dynamics in the direction of the Hessian of a function.