SYMMETRY AND SHAPE · Symmetry and shape
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2007-03-01 → 2008-12-31
- EU contribution
- €131,692
- Participants
- 1
- Scheme
- EIF
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Results in brief
Final Activity Report Summary - SYMMETRY AND SHAPE (Symmetry and shape)
Our project is devoted to the study of subsets of spaces with a high degree of symmetry. In the context which we are working in (Riemannian geometry) it is understood that an object has a high degree of symmetry if it is invariant by the action of certain group of isometries (which are the transformations of Riemannian manifolds that leave this geometric structure invariant). Such an object is called a homogeneous submanifold. The main purpose of this project has been to classify these spaces, calculate their shape and determine to what extent this shape is characteristic of them. The main result of our project, however, is concerned with the first question above. In doing so we moved to the study of more general spaces, the so-called symmetric spaces of noncompact type (many of these questions have already been addressed in the compact case). These spaces are very important in Riemannian geometry and their study goes back to the work by É. Cartan. Among all isometric actions that can be studied on a manifold, the so called hyperpolar actions have been of particular relevance over the last few years. This question is in general very difficult to handle, so we devoted our attention to a particular situation: the case when such a hyperpolar action induces a foliation on the manifold, roughly speaking, when all the homogeneous submanifolds induced by the action (the orbits of the action) have the same dimension. The main result obtained during this fellowship is the classification of all homogeneous submanifolds arising as the orbits of hyperpolar actions inducing a foliation on a symmetric space of non-compact type (J. Berndt, J. C. Díaz-Ramos and H. Tamaru, preprint arXiv:math.DG/08073517).
Data: CORDIS, © European Union
Project objective
The increasingly demanding challenges in our society and environment constantly require the development of new 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 major aim of this research project is to investigate geometric structures from the viewpoint of their symmetries, and to characterize them by some distinguished features. More precisely, we plan to investigate the shape of submanifolds of spaces equipped with a Riemannian structure that are invariant under a subgroup of the isometry group of the space. A particular objective is to study cohomogeneity one actions on symmetric spaces and characterize their orbits by geometric data. This will be achieved by using techniques recently developed by the host group and by implementing a computer program of symbolic calculus, a matter in which the applicant is experienced.This proposal suits the objectives of this action: it promotes interdisciplinary relations (cohomogeneity one 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 (the candidate comes from a less-favoured region and is hosted by a group with many international research collaborators) and improves the expertise of the applicant ensuring a position of professional maturity (the applicant is introduced in a topic of current interest, which will allow him to pioneer the investigation of this topic in his country of origin).
Original text from CORDIS.
Participants
- UNIVERSITY COLLEGE CORK, NATIONAL UNIVERSITY OF IRELAND, CORK · CORKCoordinatorIreland
Links
Data: CORDIS, © European Union