The increasingly demanding challenges in our society require the development of innovative ideas. These ideas often originate from mathematical models of phenomena in natural and life sciences, in medicine and business, in technology and engineering, in manufacturing and design, and in other areas. Many of these models lead to geometric structures describing the phenomena precisely or, in most cases, approximately. Innovation and scientific development depends upon a thorough understanding of these geometric structures.The aim of this project is to investigate geometric structures from the viewpoint of their symmetries, and to characterize them by some distinguished features. We are interested in the geometry of those submanifolds that arise as orbits of isometric actions on Riemannian manifolds. More precisely, we plan to investigate the following problems:- Classification of isometric actions on Riemannian manifolds. We will focus on polar and hyperpolar actions, or cohomogeneity one actions, that have attracted much attention over the last few years.- Study of the geometry of the orbits of isometric actions on Riemannian manifolds.- Characterization of the orbits of isometric actions by means of their geometric properties.The objectives of this proposal will be achieved by using techniques recently developed by the applicant and his previous host group and by developing a computer program of symbolic calculus, a matter in which the applicant is experienced.This proposal suits the objectives of the action: it promotes interdisciplinary relations (isometric actions have successfully been used for the construction and investigation of special structures which have substantial interest in physics; we use programming techniques to handle symbolic calculations), reinforces international relations and provides the applicant with the chance of a lasting professional integration in a research institution.