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Special invariant metrics on Lie groups and their compact quotients

Abstract

This project aims to support the Responsible Researcher in the development of research in differential geometry, facilitating her inclusion in the Brazilian mathematical community and contributing to maintain, and further strengthen, her collaborations with researchers abroad. It is expected to bring a network of international exchange to the well established relation between FAPESP and UNICAMP with research groups of excellence in Europe and Latin America.More precisely, this research proposal is aimed at the study of special geometric structures on Lie groups and their compact quotients, when they exist. The general objective of the project is to determine algebraic and topological constraints on a Lie group to carry {\em special} Riemannian conformal structures and pseudo-Riemannian metrics. Different types of {\em special} structures and metrics will be considered, namely, conformal Killing tensors, Weyl-Einstein structures, both in the Riemannian context, and ad-invariant and Ricci-flat metrics, in the pseudo-Riemannian context. In differential geometry, and especially after Milnor's work in 1976, Lie groups endowed with left-invariant metrics, constitute a source of (counter)examples to the most varied problems in geometry. This because these are manifolds with a manageable geometry, since they allow the use of Lie theoretical techniques. This relevance of Lie groups in differential geometry motivates the present research proposal. (AU)

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