Special Riemannian Geometries

(from the candidature)
Riemannian geometry lies at the heart of modern mathematics, having strong roots in all of its fields. “Special Riemannian Geometries” designates a subfield which is object of great interest nowadays and keeps plenty of interactions with Physics. It covers all questions related with Riemannian
structures (positive definite metric) characterized by some subgroup of the Special Ortogonal Lie group SO(m) or the Spin group Spin(m), where m is the dimension of some Riemannian manifold. We use those words in our title to highlight the interplay of three very active lines of research, pursued in renowned world geometry centres and to which we believe we may give quite a strong contribution. The lines are: geometry with torsion, calibrated geometry and (moduli of) instantons. In the mathematical challenges we want to give new breadth and vision, we shall be, firstly, continuing the successful efforts of a previous project with two of the researchers, and secondly, cooperating towards some major breakthroughs, counting on new input of a recent PhD in geometries with torsion and moduli spaces.

The researchers Isabel Salavessa and Rui Albuquerque have worked together in the FCT project "Geometry of Special Holonomy" (FCT/POCI/60671/2004), which was considered as “very good” by the evaluation committee and thence funded only for 50% of its initial budget. They
published several works, surpassing the goals of their project. The international cooperation was duly accomplished, having led them to the third and youngest member of the present project team.

Ana Cristina Ferreira is a recent PhD graduate from the University of Oxford, UK. She is a promising young mathematician, a former MSc student from the University of Porto where she developed her interest in Geometry with Professor Peter Gothen. In Oxford, she was a student of the acclaimed
geometer Nigel Hitchin. Her study focused essentially on geometry with torsion, but also on spin geometry and the Dirac operator, which are classical. She worked also on the geometry of 4-manifolds and became interested in the new theory of generalized geometries.

Rui Albuquerque has found a new geometry, now coined geometry of “gwistor” space. It is a remarkable general construction of a (Lie group) G2 structure on the unit sphere bundle of a Riemannian 4-manifold. The problems it has raised are far too many for one researcher willing to go further in his studies. Even if we account the collaboration of his present PhD student M. I. Magalhães Colaço.

Isabel Salavessa is a Principal Investigator in Geometry who will be bringing the input of the geometry and analysis of calibrated submanifolds in special structures. [...]