Homogeneous Einstein metrics and their geometric properties

• Applicant: Professorin Dr. Ilka Agricola
• Subject Area: Mathematics
• Term: from 2012 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 219342577

The scientific focus of this project is on the relation between geometric structures on homogeneous Riemannian manifolds and special metrics that they may carry, in particular, Einstein metrics. Einstein metrics are vacuum solutions of the Einstein equation of general relativity with a cosmological constant. The geometric structures we have in mind are compact homogeneous manifolds with special holonomy properties (both Riemannian and for more general metric connections with torsion), contact structures, almost Hermitian and paracomplex structures, G2 and hyper-K¨ahler structures as well as their relatives, a by now well established field of differential geometry with many links to mathematical physics (in particular, superstring theory). These structures occur in a natural way for solutions of spinorial field equations or other, geometrically motivated differential equations. We plan a systematic construction of new homogeneous Einstein metrics and the investigation of their weak holonomy and spinorial properties like eigenvalues of the Dirac operator. Although the scientific setting is of general geometric interest, our methods are deeply rooted in representation theory.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1388: Representation Theory