Solitons de Ricci et quelques déformations Ricci solitons and certain deformation

Abstract

This work was completed to obtain a Doctorate in Mathematics, specializing in Differential Geometry at the University of Mascara. The title of the dissertation is Ricci-Soliton and its Transformations and it comprises two significant keywords. The first, Ricci-Soliton, refers to the geometric structures that form a self-similar solution to the Ricci flow, a topic that has been extensively researched by mathematicians since Grigori Perel- man used the Ricci flow to solve the Poincare conjecture. By transformations, we mean metric changes, which are crucial tools for constructing examples of new theoretical structures in geometry and studying their rigidity. The purpose of this dissertation is to investigate the transformation and rigidity of Ricci-Solitons through various metric transformations within the framework of Riemannian geometry. Our first published results delve into the investigation of D-isometric trans- formations on a class of Riemannian manifolds and how these transformations affect the soliton structure on them. The second result provides a detailed examination of almost contact metric manifolds of class C12, termed corner manifolds by Professor Gherici Beld- jilali in his research. We study their geometric properties and solitons, as well as various deformations such as η-homothetic and ω-homothetic deformations. Additionally, we present results on their structure related to Lorentzian C12.