ON THE RICCI SOLITON EQUATION - PART I. CNOIDAL METHOD

Abstract

A smooth vector field on a Riemannian manifold is a Ricci soliton if it satisfies the Ricci soliton equation This paper presents the cnoidal method, a suitable method to obtain the Ricci solitons. The paper presents some properties of the linear and nonlinear waves. The study of Ricci solitons is related to the Tzitzeica surface and Tzitzeica equations which influence the topology of the Riemannian manifold and also influence on its geometry. Hamilton proved in [1] that the only closed gradient shrinking Ricci solitons in 2D are Einstein. In 3D, Ivey proved that all compact gradient shrinking Ricci solitons have constant positive curvature. Bohm and Wilking’s result in [2] implies the compact gradient shrinking Ricci solitons with positive curvature operator in any dimension have the constant curvature, generalizing Ivey’s result.