The curvatures of a Riemannian manifold and applications

This thesis has four main sections. In the first section main concepts have been represented in order to be comprehended well.In the second section firstly principal curvatures in a random point of a surface have been presented by being explained the origins of the surfaces in E³ geometry then included the geometrical interpretations of principal curvatures ,afterwards presented Gauss and mean curvatures of a surface and theorems that contain geometrical interpretations of these curvatures.Then Riemannian curvature tensor ,sectional curvatures,ricci curvatures and skalar curvature concepts have been taken into consideration in a detailed way and relations among these titles have been designated. And the section has been finished by examining Riemannian submanifold curvature concept with its defferent sides.In the third section Gauss? Egregium theorem and the applications of this theorem have been presented and then curvature forms and Cartan structure equations have been explained.In the last section different kinds of examples from concrete usages of this curvature have been included.KEY WORDS: Principal curvatures,Gauss curvature,mean curvature,Riemannian curvature tensor,isometry,sectional curvature,ricci curvature,skalar curvature,submanifold